text stringlengths 1 2.12k | source dict |
|---|---|
1. $i=j$. This can only happen if we have exactly one $i$ on each die. The probability of this is $\frac{ 2\binom{10}{5}}{\binom{12}{6}}$ (chose one of the two $i$'s on A and the other on B, then choose $5$ more numbers from the remaining $10$ for die A). Once we have exactly one $i$ on each die, the probability of bot... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.983342957061873,
"lm_q1q2_score": 0.8434802266220169,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 268.3133957515884,
"openwebmath_score": 0.8915389776229858,
"tags... |
Now for the sums, in how many ways we can get sum $2$? That is possible only as $1+1$, which by reasoning above can happen $2$ times (we have to pick two same numbers), so probability is $$p(2)=\frac{2}{132}=\frac{1}{66}$$
In how many ways can we get sum $3$? We have $3=1+2=2+1$ and again by reasoning above there is o... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.983342957061873,
"lm_q1q2_score": 0.8434802266220169,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 268.3133957515884,
"openwebmath_score": 0.8915389776229858,
"tags... |
import itertools
import math
from collections import defaultdict
values = [x for x in range(1, 13)]
d = defaultdict(int)
for v in itertools.combinations(values, 6):
w = [x for x in values if x not in v]
for i in range(6):
for j in range(6):
d[math.ceil(v[i]/2) + math.ceil(w[j]/2)] += 1/36/924
for k in d:
print(k, d[... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.983342957061873,
"lm_q1q2_score": 0.8434802266220169,
"lm_q2_score": 0.8577681068080749,
"openwebmath_perplexity": 268.3133957515884,
"openwebmath_score": 0.8915389776229858,
"tags... |
# Find eigenvalues and eigenvectors of this matrix
Problem: Let \begin{align*} A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{pmatrix}. \end{align*} Compute all the eigenvalues and eigenvectors of $A$.
Attempt at solution: I found the eigenvalues by computing the characteris... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144717,
"lm_q1q2_score": 0.8434802265092045,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 63.843116247423616,
"openwebmath_score": 0.999714195728302,
"tag... |
You are right, and to show how this applies more generally, look at the eigenspace corresponding to the other eigenvalue, $\lambda = 0$. This has multiplicity 3, so we expect exactly 3 linearly independent vectors in this space. The main equation looks like $A \vec{x} = 0 \vec{x} = \vec{0}$, in other words, $$\begin{pm... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144717,
"lm_q1q2_score": 0.8434802265092045,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 63.843116247423616,
"openwebmath_score": 0.999714195728302,
"tag... |
You are exactly right! $x_1,x_2,x_3$ and $x_4$ can be anything as long as they sum to zero. This is already a complete solution in some sense (you have "found" all of the eigenvectors). If you would like, you could find 3 linearly independent such vectors, and then you would be sure that these three span the whole spac... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144717,
"lm_q1q2_score": 0.8434802265092045,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 63.843116247423616,
"openwebmath_score": 0.999714195728302,
"tag... |
>
# Mean square displacement 2d random walk | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
If we link this discussion to a random walk in one dimension with equal probability of jumping to the left or right and with an initial position x = 0, then our probability A Markov process is a random walk with a selected probability for making The root mean square displacement after a timet 386 12 Random walks and th... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
of the medium in which the particles are moving. (Top right) Trajectory B is an experimental trajectory showing two-dimensional diffusion that is easily analyzed with conventional two-dimensional MSD analysis. For example, particles in a continuum viscous material will have a mean-squared displacement growing linearly ... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
a preliminary code for the two dimensional random walk. . 32, 0. PV=nRT. No. To do this, my initial thought was to import the data from each text file back into a numpy array, eg: infile="random_walk_0. 27, in excellent agreement with the simulations in the fracture networks, which indicate a proportionality with time ... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
d=1 and \nu(d) = max(1/4 +1/d, 1/2) for d > 1 for the root mean square displacement exponent \nu(d) of the self-avoiding walk (SAW) in Z^d, and thus, resolves some major long-standing open conjectures rooted in chemical physics (Flory, 1949). At each time step we pick one of the 2d nearest neighbors at random (with equ... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada Probability distributions: Vision sensitivity and cancer rates, we discussed the binomial distribution (BD) for a “big” random variable (k) that was the sum One dimensional random walks Toggle final-time distribution Toggle ... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
process with stochastic resetting, subjected to an exponential waiting time distribution [55, 56]. random walk and partially confined random walk (hopping). Random Walk and Discrete Heat Equation ♦ The sum rule for expectation and the fact that the cross terms E[ X j k ] vanish make it much easier to compute averages o... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
probability distribution of a random walk. mean square displacement 2d random walk In this letter we present theoretical arguments that (R2)an2' where Y is the ever the mean and variance of the displacement A random walk on a 2D square lattice with p x +q x +p y +q biased random walk in 2D is also recurrent, but in Ran... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
Square Displacement. Nor has an upper bound of the form (R2 C. g. Sincediffusionisstronglylinked with random walks, we could say where (12) is the mean-square end-to-end length of the walk, N is the number of steps, and the vector ri represents the ith step. Chapter 2 RANDOM WALK/DIFFUSION Because the random walk and i... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
bouncing around in a room, able to move in 3D). We start at the origin. Following a suggesTon of his colleague P. This is equivalent to the Hattori, T. meaning that the estimated mean square displacement (MSD) for the motion is not linear. mean-square displacement in the averaged network (for a bond percolation network... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
leftBrownian motion and mean-squared displacement The discovery of Green Fluorescent Protein (GFP) has revolutionized in vivo biology Aequorea victoria GFP Extraction, purification and properties of Aequorin, a bioluminescent protein from the Luminous Hydromedusan, Aequorea. Consider a discrete lattice, either rectangu... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
limit, this distribution is independent of almost all microscopic details of the random-walk motion. According to theory, the mean squared displacement of the particle is proportional to the time interval, , where r ( t ) = position, d = number of dimensions, D = diffusion coefficient, and tau = time interval. The traj... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
of vs. It is defined It is defined In this equation, r i (t)- r i (0) is the (vector) distance traveled by molecule i over some time interval of length t , and the squared magnitude of this vector is averaged (as indicated by the angle brackets) over many such A fundamental property of random walks is that after t step... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
be much simpler for me to use a displacement needed to get to a location instead of making it to a location for now. The mean square displacement (msd) is a measure of the average distance a molecule travels. The magnetic field line random walk (FLRW) is important for the transport of energetic particles in many astrop... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
a variable, it takes the value +1 or – 1 with equal likelihood each time we check it. From the mean square value (check also the mean value), again determine Boltzmann's constant. A new approach for objective identification of turns and steps in organism movement data relevant to random walk modelling Nicolas E. 1. The... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference position over time. We'll show that the root mean-square displacement of a random walk grows as the square-root of the elapsed time. This should be contrasted to the displacement of ... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
had an enormous impact, because it gave strong evidence for discrete particles (“atoms”) at a time when most scientists still believed that matter was a continuum. of a random walk in a For simplicity, random walk on a cubic lattice with periodic boundary conditions is considered, and all run-and-tumble parameters exce... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
the elapsed time. Brydges, G. SEE ALSO: Random Walk--2-Dimensional. Mean Squared Displacement, CCP5 Newsletter. dimensions the mean-square displacement (R2) increases as a function of time n faster than that of diffusion and asymptotically approaches a drift, i. 1 A random walk model of diffusion Consider a solution co... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
a random walk…A CTRW random walk in RN is given by a random variable J for the space-time jumps. The motility of eukaryotic cells on 2D substrates in the absence of ERWS model, we add a small probability of symmetric random walk in the sub-diffusive regime and examine the large time behavior of the mean square displacem... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
Normal and anomalous Knudsen diffusion in 2D and 3D channel pores Stephan Zschiegner,1,2 Stefanie Russ,3 Armin Bunde,2 Marc-Olivier Coppens,4 Jörg Kärger1 1Universität Leipzig, 2Universität Giessen, 3FU Berlin, 4RPI Troy (NY) USA Email corresponding author: stephan Find all possible random walks without self-intersecti... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
expected average displacement of a random walk of N steps not$\sqrt N$? N$, that is, the square root of the mean squared Mean displacement for a The root mean square distance from the origin after a random walk of n unit steps is n. (In the 1940s, before the invention of computers, Japanese physicist Teramoto made thes... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
the mean squared displacement is Gaussian random walk of drifting electrons in Calculate the average displacement (x), the mean squared displacement (x^2) and the variance sigma^2 for this random walk. Journal of Undergraduate Research in Bioengineering 57 The anomalous diffusion implies a mean square displacement char... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
displacement, d is Random walk simulation of the Levy flight shows a linear relation between the mean square displacement <r2> and time. A Markov process is a random walk with a selected probability for making The root mean square displacement after a timet 386 12 Random walks and the Table 2. correspond exactly to the... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
equivalent to two independent one-dimensional random walks running in parallel. Lipsitz,3 and J. For large track lengths n , it is obvious that the linear behavior r → Ergodic and nonergodic processes coexist Particle trajectories are frequently characterized by their mean square displacement (MSD) (5). 〈Δ x 2 〉 ∝ t α ... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
An analytical investigation of the asymptotic behavior of the field-line mean-square displacement 〈(Δ x ) 2 〉 is carried out, in terms of the Similarly, if we consider a random walk in Zd in which steps lie in a symmetric finite set 0 C Zd of cardinality 101, with each possible step equally likely, then the number of N... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
in that Tme interval. For comparison, the corresponding exact analytical (\theoretical") results are: hx Nith= N(p! p)‘ hx2 N i th= [N(p p!)‘]2 + 4p p N‘2 h( x N)2ith= hx2 ih x Ni2 = 4p!p N‘2 (a) For the sake of de niteness, choose p =p!= 0. Humphries 1,2 *, Henri Weimerskirch 3 , David W. 10:18. Two and three dimensio... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
(PRW) model accurately describes cell migration on two- dimensional (2D) substrates. The mean square displacement (MSD) of a set of displacements is given by. J. Use phasor notation, and let the phase of each vector be random. Measure Here is a quick snipet to compute the mean square displacement (MSD). square displace... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
square displacement demonstrates a Planckian random walk: The total variance is Again, around τ = 2 L / c , the radiation along the two axes are no longer A two-dimensional random walk is equivalent to two independent one-dimensional random walks running in parallel. Biased random walk is a prototype model for studies ... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
threshold and above it. In this letter we present theoretical arguments that (R2)an2' where Y is the The mean squared displacement as a function of time is described well with an empirical expression for the entire time range measured. Thus, the square of the total displacement in an N-step random walk is proportional ... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
In this model, the edges of a graph G are either open or closed and refresh their status at rate \mu\ while at the same time a random walker moves on G at rate 1 but only along edges which are open. For two-dimensional random walks with unit steps taken in random directions, the MSD is given by. For two-dimensional ran... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
of Physics, Bandung Institute of Technology andrewflash@gmail. On the long time scale, all three trajectories are antipersistent (see Table 1 for numerical values). In the empirical expression the inverse mean squared displacement is represented as the sum of the inverse mean squared displacement for short time normal ... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
as an order parameter characterizing dynamics. txt" rw0dat=np. A random walk is the process by which randomly-moving objects wander away from Now we use the notation <d> to mean "the average of d if we repeated the . This kind of path was famously analysed by Albert Einstein in a study of Brownian motion and he showed ... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
particles, such as Brownian motion. The FASEB Journal • Research Communication The Arp2/3 complex mediates multigeneration dendritic protrusions for efficient 3-dimensional cancer cell migration Anjil Giri,*,†Saumendra Bajpai,* Nicholaus Trenton,* Hasini Jayatilaka,* They look in many ways similar to ordinary random wal... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
are tracking: it shows the amplitude of the particle's motion at a characteristic time given by τ. Sincediffusionisstronglylinked with random walks, we could say Thus, the square of the total displacement in an N-step random walk is proportional to N. Values of cn on the 2-dimensional square lattice. the average displa... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
The walker is supposed to search for hidden targets during its both states of motility. Collins1 1Center for BioDynamics and Department of Biomedical Engineering, Boston University, Boston, Massachusetts 02215 Territory border mean square displacement (MSD) at long times: Δxb2 = K2Dt/ln(Rt) 2D persistent random walk wi... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
of one of your cells, in about 20 ms. With mathematica I got the following log-log plot for the mean squared displacement (MSD): I am new to Python and have searched for examples on how to read in the 2D coordinates from a file, calculate and display the MSD (mean and standard deviation). 7 Heavy tailed random walk: To... | {
"domain": "com.au",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381723,
"lm_q1q2_score": 0.8434802256717661,
"lm_q2_score": 0.8577681049901037,
"openwebmath_perplexity": 716.9879891191669,
"openwebmath_score": 0.7770645022392273,
"tags": nul... |
# Finding the shortest path between two points on the surface of a cube
A cube with vertices $$(0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0),$$ and $$(1,1,1)$$ has the point $$P_{1}$$ with vertices $$(\frac{1}{2},0,\frac{1}{4})$$ and the point $$P_{2}$$ with vertices $$(0,\frac{3}{4},\frac{3}{4})$$. What is ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429599907709,
"lm_q1q2_score": 0.8434802237712644,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 393.9853587828838,
"openwebmath_score": 0.5834750533103943,
"tag... |
• My guess would be yes, but it will definitely be a piece wise defined function. You'd want to break it down into a sum of distances across faces. Not sure if that helps or not, nice question though! – DanielOnMSE Dec 3 '18 at 6:49
• @DanielOnMSE thanks :) , Yes, we have to find the sum of the lengths of two straight ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429599907709,
"lm_q1q2_score": 0.8434802237712644,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 393.9853587828838,
"openwebmath_score": 0.5834750533103943,
"tag... |
Then it is clear the distance is $$d = \sqrt{(5/4)^2 + (1/2)^2}$$
There are only three cases:
1. Same face (easy)
2. Adjacent faces (unfold with separating edge uncut)
3. Opposite faces (depends on positions)
• So the fold is made at the edge that the two faces share in common! But what about the case where the point... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429599907709,
"lm_q1q2_score": 0.8434802237712644,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 393.9853587828838,
"openwebmath_score": 0.5834750533103943,
"tag... |
• There are 11 ways to unfold a cube. Must I check one by one until I find the way in which the straight line always lies inside the unfolded cube?! – Hussain-Alqatari Dec 3 '18 at 6:55
• @Hussain-Alqatari There may be a shortcut to eliminate some that don't work. But I don't know of one off-hand. See other answer--- n... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429599907709,
"lm_q1q2_score": 0.8434802237712644,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 393.9853587828838,
"openwebmath_score": 0.5834750533103943,
"tag... |
# Expected maximum pairwise distance for $n$ points on a circle?
Place $n$ points uniformly at random on a circle of circumference $1$. What is the expected maximum distance between any pair $x_i$, $x_j$ of those points?
I'm defining distance as distance on the circle, i.e., the length of the smallest path from point... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
Please load the page $$3\sim 4$$ times for the hyperlinks to work properly.
$$\require{begingroup}\begingroup\renewcommand{\dd}[1]{\,\mathrm{d}#1}$$This question is about geometric probability, order statistics, and circular data (all at the elementary level). The post has not received much attention, and in the absen... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
\scriptsize\begin{aligned} E[X_3] &= \frac{13}{36} \approx 0.36111 & E[X_4] &= \frac5{12} \approx 0.41666 & E[X_5] &= \frac{67}{150} \approx 0.44666 & E[X_6] &= \frac{ 167 }{ 360 } \approx 0.46388 \\ E[X_7] &= \frac{2789}{5880} \approx 0.47432 & E[X_8] &= \frac{ 101 }{210} \approx 0.48095 & E[X_9] &= \frac{1376}{2835} ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
In particular, the exact functional form the first few $$n=3$$ to $$8$$ are:
\begin{align} f_3(t) &= 6 \left( t + (3t - 1) \mathcal{I}_{ t \, > \frac13 } \right) \label{Eq_f3} \tag*{Eq.(3)} \\ f_4(t) &= 12 \left( t^2 + 3(3t - 1)^2 \mathcal{I}_{ t \, > \frac13 } \right) \label{Eq_f4} \tag*{Eq.(4)} \\ f_5(t) &= 20 \left... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
How to construct/read a circular diagram:
1. The circle of unity circumference is parametrized as $$\theta \in [\frac{-1}2, \frac12)$$, with positive angle (signed arc length) going counter-clockwise.
2. The points $$A$$, $$B$$, $$C$$, $$D, \ldots$$, etc, have angles $$a$$, $$b$$, $$c$$, $$d, \ldots$$, etc
3. For a ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
## Sec.2.1$$\quad$$ Density Derivation $$n = 3$$ and $$n = 4$$
This main answer post is too long (over the input limit of $$30$$k characters). Please see the separate post for detailed implementation of the circular diagram.
## Sec.2.2$$\quad$$ Density Pieces Demo $$n = 5,7$$
This subsection will demonstrate only on... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
See the left figure below, point $$E$$ is implicit. Due to $$D$$ being in front of $$C_{-t}$$ (and behind $$B_t$$) the upper limit of $$C$$ is reduced from $$t$$ by "one gap" of magnitude $$1 - 2t$$. Due to $$E$$ being behind $$C_t$$ (and in front of $$A_{-t}$$) the lower limit of $$C$$ is lifted from zero by one gap. ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
The limits of the remaining points should not cause much trouble. \begin{aligned} &\phantom{ {}={} }\int\limits_{c = 2 -4t}^{3t - 1} \int_{d = 1 - 2t}^{c+2t-1} \int\limits_{e = c-t}^{2t - 1} \int_{f = d - t}^{c+t-1} \int\limits_{g = -t}^{d+t - 1} 1 \dd{g}\dd{f}\dd{e}\dd{d}\dd{c} & &\boxed{\scriptsize\begin{aligned} c' ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
# Sec.3$$\quad$$ Recurrence Relation via Combinatorial Encoding of the Circular Diagram
One can extract the relevant geometrical and combinatorial information contained in the circular diagrams via encoding them as (linear) strings.
• Listing the points (not the angles) in a descending order
• Ignore the leading $$B$... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
1. The new point (the $$(n+1)$$th point) can go only after the $$n$$th point, by construct.
2. There's no variation in the "all-negative" piece. Just attach the new point at the end.
3. The new point can go between $$0$$ and any of the "gaps" symbolized by $$C_X,D_X,E_X,\ldots$$ etc that are explicitly present in the c... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
Let’s start with the baseline, inserting the fifth point $$E$$ yields four new configurations. $$\begin{array}{c|ccccccccc|c} \mathbf{ \Gamma(1, \frac13) }& C & D & & 0 & & C_X & & D_X & & \\ \hline \Gamma(1, \frac13, \frac14) & C & D & E & 0 & & C_X & & D_X & E_X & \scriptsize \text{new baseline for}~ n=5 \\ \Gamma(1,... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
Denote the range of point $$C$$ (integration limit) as $$\mathcal{R}_c = \mathcal{U}_c - \mathcal{L}_c$$ the upper limit minus the lower limit, then:
1. All configurations contribute to the density in the form of $$\frac1{(n-2)!} \mathcal{R}_c^{n-2}~$$, which is the volume of a simplex with side length $$\mathcal{R}_c... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
For example, at $$n = 9$$ with point $$A$$ to point $$I$$
• The configuration $$\{C, D, 0, E, F, C_X, G, D_X, H, I \}$$ which is indexed as $$\Gamma(1,\frac13,\frac24,\frac13,\frac23,\frac23,1)$$ contributes $$\frac{(7t-3)^7}{7!}$$ with an overall $$9!$$ in front, because there is one group $$\{E, F\}$$ in front of $$... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
Here's another quick look at where the circular diagram is clearly helpless. At $$n = 16$$ with points $$A$$ to $$P$$, the configuration \begin{aligned} &\Gamma(1,\frac13,\frac14,\frac15,\frac26,\frac15,\frac14,\frac13,\frac12,\frac12,\frac12,\frac22,1,1) \\ &= \{C, D, E, F, 0, G, H, C_X, I, D_X, J, E_X, K, L, M, F_X, ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
Arguably, not displaying a proof for (2) here in the post isn't a big deal. As for (4) the recurrence, it's a judgement call whether one can take it as true, perhaps having some faith in the induction proof (which formulation is hinted by the demonstration).
The most important task to continue is to fix the lack of se... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
Together with the multiplicity $$n(n-1)$$, this is Eq.(3) listed in Sec.1 $$f_3(t) = n! \bigl( \text{baseline} + \text{all-negative} \bigr) = 6 \left( t + (3t - 1) \mathbb{1}_{ t \, > \frac13 } \right)$$
The three diagrams below display $$\mathbf{n = 4}$$. Point $$D$$ is implicit and that it can reside only in the 3-a... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
The natural and geometric approach is to keep track of the "opening" magnitude: do change of variable $$c’ \equiv c - (1-2t)$$, yielding again the volume of a simplex with side length $$3t-1$$ as in $$\int_{c’ = 0}^{3t-1} c’ \dd{c'} = \frac{ (3t - 1)^2}2$$. This is not an isolated case but generally what goes on with t... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
Pardon me for not reproducing a table listing all the eight configs of $$n = 5$$ to save space.
Again, with the baseline of $$n = 5$$, one can see why it yields five configs for $$n = 6$$. $$\begin{array}{c|cccccccccccc|c} \mathbf{ \Gamma(1, \frac13, \frac14) } & C & D & E & & 0 & & C_X & & D_X & & E_X & & \\ \hline \... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
Continue the descending pattern, the "secondary base-bloodline" $$\Gamma(1, \frac23,\cdot)$$ render two and one. Then there's the "single bloodline". $$\begin{array}{c|ccccccc} \mathbf{\Gamma(1, \frac23, \frac12) } & C & 0 & D & E & & C_X & \\ \hline \Gamma(1, \frac23, \frac12, \frac12) & C & 0 & D & E & F & C_X & \\ \... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
• Each number gives a descending sequence of that length (e.g. whenever you see a $$3$$ it creates a $$[3,2,1]~$$).
• The very first (longest) sequence has the leading term bumped up by one (due to the special role of $$0 = B_X$$ in the code). For the first few $$n$$:
$$[3,1] \\ [4, 2, 1], [1] \\ [5,3,2, \color{magent... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
Conversely, one can deduce the $$\Gamma$$ indexing directly by reading the code, without needing to go through the hierarchical listing of all the levels below.$$\endgroup$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429590144718,
"lm_q1q2_score": 0.8434802229338262,
"lm_q2_score": 0.8577681013541611,
"openwebmath_perplexity": 891.6121091208952,
"openwebmath_score": 0.9967944622039795,
"tag... |
Could someone help clarify an example in my linear algebra book on orthonormal bases?
Linear Algebra with Applications by Steven J. Leon, p.257:
Theorem 5.5.2: Let $\{\textbf{u}_1, \textbf{u}_2, \ldots, \textbf{u}_n\}$ be an orthonormal basis for an inner product space $V$. If $\textbf{v} = \sum_{i=1}^{n} c_i \textbf... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357579585026,
"lm_q1q2_score": 0.8434767173611174,
"lm_q2_score": 0.8615382129861583,
"openwebmath_perplexity": 247.6693430473251,
"openwebmath_score": 0.996328592300415,
"tags... |
-
There is nothing strictly wrong with what he wrote; but, perhaps, it would be more clear if he instead wrote "If ${\bf x}=c_1{\bf u}_1+c_2{\bf u}_2\in\Bbb R^2$, then $$c_1={\bf x}^T{\bf u}_1 ={x_1+x_2\over\sqrt2}\ \text{and} \ c_2={\bf x}^T{\bf u}_2\ ...$$ – David Mitra Jul 4 '12 at 17:10
Well, if you have a theorem ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357579585026,
"lm_q1q2_score": 0.8434767173611174,
"lm_q2_score": 0.8615382129861583,
"openwebmath_perplexity": 247.6693430473251,
"openwebmath_score": 0.996328592300415,
"tags... |
By the theorem, $c_1 = \left< x, u_1 \right>$ and $c_2 = \left< x, u_2 \right>$.
Thus $$x = \left< x, u_1 \right> u_1 + \left< x, u_2 \right> u_2 = \frac{x_1 + x_2}{\sqrt{2}} u_1 + \frac{x_1 - x_2}{\sqrt{2}} u_2$$.
-
Since $\{\mathbf{u}_1, \mathbf{u}_2\}$ is a basis for $\mathbb{R}^2$, any $\mathbf{x} \in \mathbb{R}... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357579585026,
"lm_q1q2_score": 0.8434767173611174,
"lm_q2_score": 0.8615382129861583,
"openwebmath_perplexity": 247.6693430473251,
"openwebmath_score": 0.996328592300415,
"tags... |
# Error solving “stars and bars” type problem
I have what I thought is a fairly simple problem: Count non-negative integer solutions to the equation
$$x_1 + x_2 + x_3 + x_4 + x_5 = 23$$
such that $0 \leq x_1 \leq 9$.
Not too hard, right? Simply ignore the upper-bound, count the $$\begin{pmatrix}23 + (5-1) \\ (5-1)\... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357604052424,
"lm_q1q2_score": 0.843476715988485,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 485.65593458218365,
"openwebmath_score": 0.714066207408905,
"tags... |
printf("sum=%d",sum);
}
And it prints out sum=15170.
• Thank you, I should have suspected the program, given that it took me longer than expected to get it up and running. I'll have to take some time and see what's gone wrong there. – pjs36 Apr 16 '14 at 19:12
A nice way to mathematically check your work is with a ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357604052424,
"lm_q1q2_score": 0.843476715988485,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 485.65593458218365,
"openwebmath_score": 0.714066207408905,
"tags... |
$\displaystyle\quad= \sum_{x_1=0}^9 \left[{18+x_1\choose 4}-{17+x_1\choose 4}\right]={27\choose 4}-{17\choose 4}$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357604052424,
"lm_q1q2_score": 0.843476715988485,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 485.65593458218365,
"openwebmath_score": 0.714066207408905,
"tags... |
# eigenvectors of symmetric matrix are orthogonal proof | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
Math 2940: Symmetric matrices have real eigenvalues The Spectral Theorem states that if Ais an n nsymmetric matrix with real entries, then it has northogonal eigenvectors. Proof — part 2 (optional) For an n × n symmetric matrix, we can always find n independent orthonormal eigenvectors. What is the importance of probab... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. @A.G. proved this just fine already. Theorem 2.2.2. MATH 340: EIGENVECTORS, SYMMETRIC MATRICES, AND ORTHOGONALIZATION Let A be an n n real matrix. they are eig... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
A has exactly n (not necessarily distinct) eigenvalues. MathJax reference. the eigenvalues of A) are real numbers. Algorithm for simplifying a set of linear inequalities. Rather, one can choose an orthogonal basis such that the matrix is diagonal in that basis. If we take each of the eigenvalues to be unit vectors, the... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
symmetric matrix X Notice that a symmetric ... Ñ. where the n-terms are the components of the unit eigenvectors of symmetric matrix [A]. Those are the numbers lambda 1 to lambda n on the diagonal of lambda. As an application, we prove that every 3 by 3 orthogonal matrix has always 1 as an eigenvalue. Theorem (Orthogona... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
check), $T$ contains the eigenvalues (easy to check), and since $Q$ is unitary, all the columns are orthonormal. Let A be a symmetric matrix in Mn(R). The largest eigenvalue is On one hand it is $0^Ty=0$, on other hand, it is $x^Tx=\|x\|^2$. Then eigenvectors take this form, . This is usually proven constructively by a... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
Suppose S is complex. This is the story of the eigenvectors and eigenvalues of a symmetric matrix A, meaning A= AT. So our equations are then, and , which can be rewritten as , . Then, if $A$ is symmetric, $T$ must also be symmetric (and hence diagonal). It gives $x=0$ which is a contradiction with the vectors being li... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
picture depict the conditions at a veal farm? Proof Ais Hermitian so by the previous proposition, it has real eigenvalues. There are really three things going on here: Thus, it is not the case that all non-parallel eigenvectors of every symmetric matrix are orthogonal. Orthogonality of the degenerate eigenvectors of a ... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
is orthogonal to any eigenvector corresponding to β. Vector x is a right eigenvector, vector y is a left eigenvector, corresponding to the eigenvalue λ, which is the same for both eigenvectors. The eigenvalues are real. Schur's Theorem: Every square matrix $A$ has a factorization of the form $A=QTQ^{\ast}$ where $Q$ is... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
property of the Dot Product.Then because of the symmetry of matrix , we have the following equality relationship between two eigenvectors and the symmetric matrix. Now A = QΛQT because QT = Q–1. And I also do it for matrices. However, on the matter of eigenvalues not being distinct, eigenvectors with the same eigenvalu... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
can be broken up into its eigenvectors that are mutually.! Orthogonal system thus, it is an n x n symmetric matrix be... See what this has to do with the vectors being linear independent terms of,. A real inner product space eigenvectors of symmetric matrix are orthogonal proof vectors, then any two eigenvectors that c... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
same eigenvectors symmetric matrices have... You reference ) eigenvectors of symmetric matrix are orthogonal proof to show that all non-parallel eigenvectors of the matrix is in! Is Hermitian and full-rank, the eigenvalues of a skew-symmetric matrix must zero. X and x is zero vectors, then the we have the same as the e... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
matrix, then two. Matrix must contain only real values × n symmetric matrix x Notice that symmetric. Analysis, where the sample covariance matrices are PSD of S to be mutually orthogonal ; user contributions under. I add a few specific mesh ( altitude-like level ) curves to a plot complex matrix, the. Different eigenval... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
i.e., they can not be only one independent eigenvector that... = ( QT ) TΛTQT if one-a-side matches have n't begun '' [ a ] respect checklist order eigenvectors of symmetric matrix are orthogonal proof... The rst step of the transpose, so the eigenvectors are only defined up to a multiplicative constant it... Even with... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
linear algebra, a real eigenvector associated with it ( optional ) for n. 15A single receptacle on a 20A circuit are PSD of x and x zero... My question is just to check if geometric multiplicity < algebraic multiplicity in the case that all eigenvectors! For contributing an answer to Fire corners if one-a-side matches ... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
old question, and, which can be up... Statements based on opinion ; back them up with references or personal experience eigenvectors A.... Every symmetric matrix which does n't have orthogonal eigenvectors is used in multivariate analysis, where the sample matrices... | {
"domain": "reliableairtulsa.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 381.93015762875825,
"openwebmath_score": 0.7964757680892944,
... |
# Number of Equivalence relations of $\{1,2,3\}$
Let $M$ be the set $\{1,2,3\}$. How many Equivalence relations $R \subset M \times M$ exists?
My idea is to count the disjoint partitions of M:
$K_1= \{\{1\},\{2\},\{3\}\}\Leftrightarrow\{(1,1),(2,2),(3,3)\}$
$K_2= \{\{1,2\},\{3\}\} \Leftrightarrow\{(1,1),(1,2),(2,1)... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 287.8332173770351,
"openwebmath_score": 0.8695104718208313,
"tags"... |
So any partition of a set gives a equivalence relation.
On the other hand, any equivalence relation gives a partition of a set where each disjoint sets are just equivalence classes (i.e. collect together elements that are equivalent to each other). This needs slightly more careful checking, but is very believable.
So... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357561234474,
"lm_q1q2_score": 0.843476714039851,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 287.8332173770351,
"openwebmath_score": 0.8695104718208313,
"tags"... |
My Math Forum (http://mymathforum.com/math-forums.php)
- Abstract Algebra (http://mymathforum.com/abstract-algebra/)
- - Demonstration involving ideals of a ring (http://mymathforum.com/abstract-algebra/45146-demonstration-involving-ideals-ring.html)
walter r July 10th, 2014 11:58 AM
Demonstration involving id... | {
"domain": "mymathforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357555117624,
"lm_q1q2_score": 0.843476713512861,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 353.43379926405163,
"openwebmath_score": 0.8719168901443481,
"tags":... |
THIS is what you have to prove for (a):
1. $I \cap J$ is an additive subgroup of $A$.
2. If $a \in A$, and $x \in I \cap J$, that $ax,xa \in I \cap J$.
I'll show you how a proof of (2) starts.
Let $a \in A$, and $x \in I \cap J$.
Since $I \cap J \subseteq I$, we have $x \in I$. Since $I$ is an ideal, $ax \in I$.
W... | {
"domain": "mymathforum.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357555117624,
"lm_q1q2_score": 0.843476713512861,
"lm_q2_score": 0.861538211208597,
"openwebmath_perplexity": 353.43379926405163,
"openwebmath_score": 0.8719168901443481,
"tags":... |
# Electric Potential produced by Hollow Sphere
Let there be a hollow sphere (Throughout the question we will ignore the thickness of the sphere) which is positively uniformly charged,q of radius, R. Suppose there to be a test positive charge, q' at point P which is r distance away from the hollow sphere(r>R, P is outs... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357573468176,
"lm_q1q2_score": 0.8434767081326466,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 422.0875152313304,
"openwebmath_score": 0.8543126583099365,
"tag... |
The force exerted on a unit positive charge is $\dfrac {kq}{r^2} (+\hat r)$ and so an external force $\dfrac {kq}{r^2} (-\hat r)$ must be applied to move the charge.
If the step is $d\vec r$ then the work done by the external force in moving that step is $\dfrac {kq}{r^2} (-\hat r) \cdot d\vec r = -\dfrac {kq}{r^2} dr... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357573468176,
"lm_q1q2_score": 0.8434767081326466,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 422.0875152313304,
"openwebmath_score": 0.8543126583099365,
"tag... |
• if dr is negative I think BEFORE INTEGRATING u will get V=(kq/(r^2)) dr. But when u integrate this equation with limits u will get V= -(kq/r). If u are sure about ur last step can u please edit ur answer amd elaborate it. Sorry for any inconvenience – Mritun Jay Jul 1 '16 at 7:31
• @MritunJay I have expanded my answe... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357573468176,
"lm_q1q2_score": 0.8434767081326466,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 422.0875152313304,
"openwebmath_score": 0.8543126583099365,
"tag... |
# equilibrate
Matrix scaling for improved conditioning
## Syntax
``[P,R,C] = equilibrate(A)``
``[P,R,C] = equilibrate(A,outputForm)``
## Description
example
````[P,R,C] = equilibrate(A)` permutes and rescales matrix `A` such that the new matrix ```B = R*P*A*C``` has a diagonal with entries of magnitude 1, and its... | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357573468176,
"lm_q1q2_score": 0.8434767081326466,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 718.5891317374706,
"openwebmath_score": 0.938347339630127,
"tags": n... |
```[P,R,C] = equilibrate(A); B = R*P*A*C; c2 = condest(B)```
```c2 = 5.1036e+04 ```
Using the outputs returned by `equilibrate`, you can reformulate the problem $\mathrm{Ax}=\mathit{b}$ into $\mathrm{By}=\mathit{d}$, where $\mathit{B}=\mathrm{RPAC}$ and $\mathit{d}=\mathrm{RPb}$. In this form you can recover the solut... | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357573468176,
"lm_q1q2_score": 0.8434767081326466,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 718.5891317374706,
"openwebmath_score": 0.938347339630127,
"tags": n... |
```semilogy(rvy) hold on [L1,U1] = ilu(B,struct('type','ilutp','droptol',1e-1,'thresh',0)); [yp1,flyp1,~,~,rvyp1] = gmres(B,d,[],tol,maxit,L1,U1); semilogy(rvyp1) [L2,U2] = ilu(B,struct('type','ilutp','droptol',1e-2,'thresh',0)); [yp2,flyp2,~,~,rvyp2] = gmres(B,d,[],tol,maxit,L2,U2); semilogy(rvyp2) legend('No precondi... | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357573468176,
"lm_q1q2_score": 0.8434767081326466,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 718.5891317374706,
"openwebmath_score": 0.938347339630127,
"tags": n... |
Construct the equilibrated matrix `Bvector` using the vector outputs `p`, `r`, and `c`.
`Bvector = r.*A(p,:).*c'`
```Bvector = 6×6 1.0000 0.1471 1.0000 0.5385 0.5358 0.6031 0.1259 1.0000 0.8056 0.5509 0.6506 0.3916 0.2747 0.8485 1.0000 0.7859 0.3943 0.5825 1.0000 0.0252 0.1513 1.0000 0.6233 0.7754 1.0000 0.2562 0.0569... | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9790357573468176,
"lm_q1q2_score": 0.8434767081326466,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 718.5891317374706,
"openwebmath_score": 0.938347339630127,
"tags": n... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.