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# How to store animation properties/values, and restore them on callback

I'm implementing a Python script that sets a start frame for multiple selected objects. The objects each have their own NLA strips with their respective values. I've been trying to 'group' these animations together, and shift them according to a user input value. For example, if a user inputs the start frame as 200, the offsets start from that frame.

Making each of the NLA strips move by +value works for adding to the NLA strips seems to work well, but when the value is changed to something lower, it would keep adding to the start frame value. Also, if the value is adjusted slightly higher, like from frame 241 to 242, the strip jumps 242 frames.

I've tried using conditional statements/expressions to control the values, but on user update, the results tend to vary or cause the animations to start on the wrong frame.

A solution I think would be to have the strips revert to their original values before each update, but I don't know how to achieve such a function. The code below is what I have so far. I have a Blend example with 3 cubes attached below

import os
import re
import bpy
from bpy.props import *
import bpy.utils.previews
from bpy.types import WindowManager
from bpy.props import StringProperty, BoolProperty, IntProperty, CollectionProperty
from bpy.types import WindowManager
from bpy.props import (StringProperty,
PointerProperty,
)
from bpy.types import (Panel,
Operator,
PropertyGroup,
)

# Panel
class PreviewsPanel(bpy.types.Panel):
# Create a Panel in the Tool Shelf
bl_category = "My Panel"
bl_label = "My Panel"
bl_space_type = 'VIEW_3D'
bl_region_type = 'TOOLS'

# Draw
def draw(self, context):
layout = self.layout
scn = context.scene
layout.label("")
row = layout.row(align=False)
box = layout.box()
row = box.row()
row.prop(context.window_manager, "slide_1_start",text = "Start")
box = layout.box()
row = box.row()

def frame_slide(self, context):
obj = bpy.context.selected_objects
ob = bpy.context.object
DEFAULT = 0
STRINGVALUE = bpy.data.window_managers["WinMan"].slide_1_start
START1 = 70

for obj in bpy.context.selected_objects:
obj.animation_data.nla_tracks[0].strips[0].frame_start += STRINGVALUE
obj.animation_data.nla_tracks[0].strips[0].frame_end += STRINGVALUE
obj.animation_data.nla_tracks[0].strips[0].frame_start += 0
obj.animation_data.nla_tracks[0].strips[0].frame_end += 0
obj.animation_data.nla_tracks[0].strips[0].scale = 1

for obj in bpy.context.selected_objects:

obj.animation_data.nla_tracks[0].strips[0].frame_start += 0
obj.animation_data.nla_tracks[0].strips[0].frame_end += 0
obj.animation_data.nla_tracks[0].strips[0].scale = 1
return None

#####################################################################

# Register
def register():
bpy.types.WindowManager.slide_1_start = bpy.props.IntProperty(name="Slide_1_SF", update=frame_slide,min=1,options={'SKIP_SAVE'})
bpy.utils.register_module(__name__)

# Unregister
def unregister():
bpy.utils.unregister_module(__name__)
bpy.utils.unregister_module(__name__)

if __name__ == '__main__':
register()

Blend file

So, I want to create a script so that the original NLA strip values will be stored. if a user sets the start frame to 24, the NLA strips shift by +24. If a user changes the start frame to 13, original NLA strips values are restored and then shifted +13.

An example of what I'm trying to achieve is basically similar to: if I have a scene of multiple objects (like dominoes or cubes), I want to set a start frame for the animation without altering their offsets from each other.

• Is there a reason why you didn't put this in an bpy.types.Operator. Perhaps the solution is to have another value to store the previous setting slide_1_start_previous subtract those two from each other and use the difference for movement of the strips – J. Bakker Mar 7 at 7:30
• It is quite hard to read your code as it has a lot of unneeded code. can you clean your code up a bit? – J. Bakker Mar 7 at 7:36

I only added the relevant parts of the code. The solution is to keep an additional setting that remembers the previous setting slide_1_start_previous. We now can calculate the difference by subtracting both values. and shift the strip using this value.

def frame_slide(self, context):
difference = self.slide_1_start - self.slide_1_start_previous

for obj in bpy.context.selected_objects:
strip = obj.animation_data.nla_tracks[0].strips[0]
scale = strip.scale
strip.frame_start += difference
strip.frame_end += difference
strip.scale = scale

self.slide_1_start_previous = self.slide_1_start
return None

# Register
def register():
bpy.types.WindowManager.slide_1_start = bpy.props.IntProperty(name="Slide_1_SF", update=frame_slide,min=1,options={'SKIP_SAVE'})
bpy.types.WindowManager.slide_1_start_previous = bpy.props.IntProperty(default=1)

bpy.utils.register_module(__name__)

As you didn't implement it as an operator it is hard to determine when you want to reset these values. I would suggest placing this in an operator.

• Using the difference was actually a concept I'd already tested - it gives a few errors for some reason. When the value is adjusted with a slider, it works well. However, when the value is manually entered in the slider, it does not shift the NLA strips properly. – Dassie Mar 7 at 14:30


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# Delta function Definition and 13 Discussions

In mathematics, the Dirac delta function (δ function) is a generalized function or distribution, a function on the space of test functions. It was introduced by physicist Paul Dirac. It is called a function, although it is not a function R → C.
It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. No function has these properties, such that the computations made by theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.
In engineering and signal processing, the delta function, also known as the unit impulse symbol, may be regarded through its Laplace transform, as coming from the boundary values of a complex analytic function of a complex variable. The convolution of a (theoretical) signal with a Dirac delta can be thought of as a stimulation that includes all frequencies. This leads to a resonance with the signal, making the theoretical signal "real" (i.e. causal). The formal rules obeyed by this function are part of the operational calculus, a standard tool kit of physics and engineering. In many applications, the Dirac delta is regarded as a kind of limit (a weak limit) of a sequence of functions having a tall spike at the origin (in theory of distributions, this is a true limit). The approximating functions of the sequence are thus "approximate" or "nascent" delta functions.

View More On Wikipedia.org
1. ### Double delta potential

Summary:: I have a problem with a particle, which gets scatterd at a double delta-potential Hello, I am really stuck with the floowing problem: A particle moves from the left along the x-axis and gets scatterd at a one-dimensional potential V(x)=a[dirac delta of x) +b [dirac delta of x-c]...
2. ### S-wave phase shift for quantum mechanical scattering

a.) The potential is a delta function, so ##V \left( r \right) = \frac {\hbar^2} {2\mu} \gamma \delta \left(r-a \right)##, therefore ##V \left( r \right) = \frac {\hbar^2} {2\mu} \gamma ## at ##r=a##, and ##V \left( r \right) = 0## otherwise. I've tried a few different approaches: 1.) In...
3. ### Problem with the Finite Element Method applied to Electrostatics

Hi! I have a code that solve the poisson equation for FEM in temperature problems. I tested the code for temperature problems and it works! Now i have to solve an Electrostatic problem. There is the mesh of my problem (img 01). At the left side of the mesh we have V=0 (potencial). There is a...
4. ### I Is the derivative of a discontinuity a delta function?

In these notes, https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/MIT8_04S16_LecNotes10.pdf, at the end of page 4, it is mentioned: (3) V(x) contains delta functions. In this case ψ'' also contains delta functions: it is proportional to the product of a...
5. ### Recovering the delta function with sin(nx)/x

Homework Statement Ultimately, I would like a expression that is the result of an integral with the sin(nx)/x function, with extra terms from the expansion. This expression would then reconstruct the delta function behaviour as n goes to infty, with the extra terms decaying to zero. I...
6. ### 2D Integrating With Quadratic Arg. of Delta Function

Homework Statement I have a 2D integral that contains a delta function: ##\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp{-((x_2-x_1)^2)+(a x_2^2+b x_1^2-c x_2+d x_1+e))}\delta(p x_1^2-q x_2^2) dx_1 dx_2##, where ##x_1## and ##x_2## are variables, and a,b,c,d,e,p and q are some real...
7. ### A Inverse Laplace transform of F(s)=exp(-as) as delta(t-a)

This is mostly a procedural question regarding how to evaluate a Bromwich integral in a case that should be simple. I'm looking at determining the inverse Laplace transform of a simple exponential F(s)=exp(-as), a>0. It is known that in this case f(t) = delta(t-a). Using the Bromwich formula...
8. ### I Square root of the delta function

Is square root of delta function a delta function again? $$\int_{-\infty}^\infty f(x) \sqrt{\delta(x-a)} dx$$ How is this integral evaluated?
9. ### Finding the Delta Function of a Thin Ring

Homework Statement [/B] A very thin plastic ring (radius R) has a constant linear charge density, and total charge Q. The ring spins at angular velocity \omega about its center (which is the origin). What is the current I, in terms of given quantities? What is the volume current density J in...
10. ### A Divergence of 1/r^n....n>0

I have read in Griffiths electrodynamics that divergence of 1/r^2 is delta function and I thought it was the only special case...I have understood the logic there... but a question came in mind...what would happen in general if the function is 1/r^n ...where n is positive integer>0...because the...
11. ### How to make a delta function signal in a circuit?

Basically I want to test my analog circuit using a forcing function that has a form of a delta function. The function generator I use outputs sine wave, triangular wave and square wave (+ve and -ve output in one period). Are there any ways to produce a square wave that has an output for like 5%...
12. ### Integral of sin and cos

Starting from FT relation of delta function, I can write the followings: $$\int_{-\infty}^{\infty} \cos{\alpha x} dx = 0$$ $$\int_{-\infty}^{\infty} \sin{\alpha x} dx = 0$$ The question is how am I supposed to prove those equations, sin and cos are stable oscillating functions.
13. ### Integrating a delta function with a spherical volume integral

Homework Statement Integrate $$\int_V \delta^3(\vec r)~ d\tau$$ over all of space by using V as a sphere of radius r centered at the origin, by having r go to infinity. Homework Equations The Attempt at a Solution This integral actually came up in a homework problem for my E&M class and...

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# How to import bibliography (from .bib file) to svg or other Inkscape friendly format

I want to make a poster in Inkscape and include the bibliography I have in .bib file. When I had only one item in the list, I just copied the text from generated pdf and manually formatted it (adding the italics etc.). But I can imagine that when there is a long list of references, copying and formatting gets tedious and error-prone.

For maths equations I convert the LaTeX code to SVG and I was wondering if the same can be done with bibliography? The downside of this solution is that the resulting SVG file won't let me format text anymore so I can't control the line breaking.

My question is - is there an easy way to generate the bibliography from .bib file and convert it to SVG or another format that will allow further text formatting?

• It's not totally clear to me what you want to achieve. On the one hand you apparently do not want to format manually, but on the other hand you to to convert to to a format that "will allow further text formatting"? – Daniel Jun 10 '13 at 19:43
• Maybe HTML is a useful intermediate format for you? That could be created easily with bib2html or similar tools. – Daniel Jun 10 '13 at 19:45
• Depending on the time that is still available and your content, why not try instead to do the poster in LaTeX, not in Inkscape? There are document classes for posters... I would try this since you seem to have LaTeX sources for the rest of the material, too. – Count Zero Jun 10 '13 at 19:46
• @Daniel I don't want to format the italics in every entry, but I want to control where the line-breaking is while I format the poster as a whole. But I see your point...;) – Rabbit Jun 10 '13 at 19:53
• @CountZero Not too much time left and I am not an expert in LaTeX. But for the next poster this would be probably the best solution. – Rabbit Jun 10 '13 at 19:57

Since Inkscape can load PDF files, first render only the bibliography exactly as you want with pdflatex (without number of page, headers, etc).

Then simply load the generated file to Inkscape. Group (Ctrl-G) all the text elements to maintain the format, copy the new object and paste in your poster.

Note that you can check an option to convert fonts names to a "similar" font available. That will change/ruin the format, but you can maintain the names and later change the font as usual in text boxes of svg files. To avoid font changes due to fonts that are unavailables for Inkscape, instead of pdflatex use XeLaTeX or LuaLaTeX with a available true type system font. A MWE using the Arial like Umpush:

\documentclass[english]{article}
\usepackage{fontspec}
\setmainfont[Ligatures=TeX]{Umpush}
\usepackage{babel}
\begin{document}
\bibliographystyle{plain}
\nocite{*}
\bibliography{borra}

\end{document}

The following example do not import well the font Arial since the "J" render a right space, but this seem only a specific font problem of this font:

But even with a more exotic font like Anaktoria the import just looks well:

• Good idea. Unfortunately after the import the fonts got really messy, some letters are overlapping and the italic type is not recognized at all - instead some other font is used. – Rabbit Jun 10 '13 at 20:37
• @Rabbit. See the updated answer. – Fran Jun 10 '13 at 21:00
• Or you can try the tool pdf2svg, it will turn the fonts into svg paths. See e.g. this answer: tex.stackexchange.com/a/74915 – matth Jun 10 '13 at 21:36
• The advantage of maintain as text instead as paths is you still can do some changes. And probably this result also in a lighter file. For a complex poster perhaps this does matter. – Fran Jun 10 '13 at 21:44


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# Math Help - Inverse Issues

1. ## Inverse Issues

Is the inverse of f^(-1) (x) specifically sin^(-1) (x) [arcsinx] or does the inverse of f^(-1) (x) apply to every trigonometric inverse?

2. The inverse of f^(-1)(x) is f(x).

Maybe I don't understand the question.

3. Maybe I copied something wrong in my calculus notes?

4. Is the inverse of f^(-1) (x) ... like TK said.. the inverse of the inverse is the original function.

...specifically sin^(-1) (x) [arcsinx] ... yes, sin^(-1)(x) is also known as arcsin(x).

...does the inverse of f^(-1) (x) apply to every trigonometric inverse? yeah... like TK said. What are you talking about??

5. Strictly speaking, the trig functions don't have inverses because they are not "one to one". That is, $sin(0)= sin(\pi)= 0$ so what should $sin^{-1}(0)$ be?. What we can do is restrict the functions: if we restrict x to be between $-\pi$ and $\pi$ then It is one to one and so has an inverse (and $sin^{-1}(0)= 0$).

The notation, $f^{-1}(x)$, can be used for the inverse of any function that has an inverse. "arcsin(x)" is a different notation that means exactly the same as " $sin^{-1}(x)$".


text:
← Back to Search

# Optical-fiber-based Measurement Of An Ultrasmall Volume High-Q Photonic Crystal Microcavity

K. Srinivasan, P. Barclay, M. Borselli, O. Painter
Published 2004 · Physics

A two-dimensional photonic crystal semiconductor microcavity with a quality factor $Q\ensuremath{\sim}40,000$ and a modal volume ${V}_{\mathrm{eff}}\ensuremath{\sim}0.9$ cubic wavelengths is demonstrated. A micron-scale optical fiber taper is used as a means to probe both the spectral and spatial properties of the cavity modes, allowing not only measurement of modal loss, but also the ability to ascertain the in-plane localization of the cavity modes. This simultaneous demonstration of high-$Q$ and ultrasmall ${V}_{\mathrm{eff}}$ in an optical microcavity is of potential interest in nonlinear optics, optoelectronics, and quantum optics, where the measured $Q$ and ${V}_{\mathrm{eff}}$ values could enable strong coupling to both atomic and quantum dot systems.
This paper references

This paper is referenced by
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# The Dedekind Eta Function in Physics

+ 12 like - 0 dislike
4247 views

This interesting little fellow popped up in some operator algebra (Witt / Virasoro Lie algebra) I was exploring, so I've been curious about where else the Dedekind $\eta$-function makes a cameo appearance and found that in physics it's related to the

1) statistical parameters of solvable Ising models

(See "The Reasonable and Unreasonable Effectiveness of Number Theory in Statistical Mechanics" by G. Andrews and "Introduction to Exactly Solvable models in Statistical Mechanics" by C. Tracy.)

The difference between the average local occupation densities of two sub-lattices of a hard hexagon model of a lattice gas given on pp. 368-371 of Tracy is $R(\tau)=\frac{n(\tau)\eta(5\tau)}{\eta^2(3\tau)}$.

2) partition functions (statistical mechanics variety) for colored bosons moving on a line (1/24 is the associated Casimir energy) and one-color fermions

3) operator traces (characters) for the infinite dimensional Lie algebras $\widehat{su}_n$, equivalent to 2-Dim current algebras

4) partition function of a microscopic black-hole in a 5-Dim D-brane

5) string theory guage corrections

(For 2-5, see "Nucleon Structure, Duality and Elliptic Theta Functions" by W. Scott. For item 2, see also "Vertex Operators and Modular Forms" by G. Mason and M. Tuite.)

From pg. 39 of "Fivebrane instantons ..." and on pg. 11 of "D3 instantons ...," a correction to the field basis (of the RR axion dual to D3-branes) in type IIB string perturbation theory related to the action of S-duality in ten dimensions:

$\tilde{c_a} \mapsto \tilde{c_a}-\tilde{c}_{2,a}\:\epsilon(g)$ where, with $g=\binom{a\:\:b}{c\:\:d}$,

$$\exp(2\pi i \epsilon(g))=\frac{\eta\left [ \frac{a\tau+b}{c\tau+d} \right ]}{\left ( c\tau+d \right )^{\frac{1}{2}}\eta(\tau)}.$$

6) partition function in 2+1 dimensions and vanishing chemical potential of non-relativistic fermions in a constant magnetic field

("Nonrelativistic Fermions in Magnetic Fields: a Quantum Field Theory Approach" by O. Espinosa, J. Gamboa, S. Lepe, and F. Mendez)

7) physics of gauge theories and the Dirac operator

(See "The Logarithm of the Dedekind $\eta$ Function" by M. Atiyah.)

Michael Atiyah even goes so far as to say, "It seems therefore timely to attempt to survey the whole development of the theory of $\log(\eta)$, putting results in their natural order and in the appropriate general context. This is the aim of the present paper, in which the emphasis will be strongly geometrical. In a sense we shall show that the latest ideas from physics [circa 1987] provide the key to a proper understanding of Dedekind's original results."

8) knots and dynamics

(See "Knots and Dynamics" by E. Ghys, and Chapter 2 A New Twist in Knot Theory in Dana MacKenzie's book What's Happening in the Mathematical Sciences Vol. 7.)

Ghys presents the equation $$24\log\eta\left(\frac{a\tau+b}{c\tau+d}\right)=24\: log(\eta(\tau))+6\: log(-(c\tau+d)^{2})+2\pi i\:\mathfrak{R}\left(\binom{a\: b}{c\: d}\right)$$

where $\mathfrak{R}$ is the Rademacher function, which he relates to the linking number

between two knots related to modular/Lorenz flow: “For every hyperbolic element $A=\binom{a\: b}{c\: d}$ in $PSL(2,Z)$, the linking number between the [modular/Lorenz] knot $k_A$ and the trefoil knot $l$ is equal to $\mathfrak{R}(A)$ ....”

9) string/brane partition functions, propagators, and metrics

In "String Theory" by S. Nibbelink, $\eta$ occurs in the denominator of string partition functions for fermionic and bosonic zero modes (pp. 163-7).

A coefficient in the 10-dim metric for a 7-brane is given as $e^{\phi}=\tau_2 \eta^2\bar{\eta}^2|\prod_{i=1}^{k}(z-z_i)^{-\frac{1}{12}}|^2$ on pg. 493 of "Supergravity vacua and solitons" by G. Gibbons.

In what other contexts in physics does the Dedekind $\eta$ function take a bow?

(Edit) Moreover, since this is a community wiki and not a test question with one best answer but an attempt to come to a better understanding of the $\eta$-function and associated math and physics, I invite people to expand on any of the items with specifics (e.g., exact formulas), more references, and/or insightful commentaries (e.g., what you believe are important aspects of the references).

Other appearances: In Gliozzi's "The Infrared Limit of QCD Effective String" on pg. 14; Panero's "A numerical study of confinement in compact QED" on pg. 4; Zahed's "Holographic Pomeron and Primordial Viscosity" on pg. 1; Caselle and Pinn's "On the Universality of Certain Non-Renormalizable Contributions in Two-Dimensional Quantum Field Theory" on pg. 3; Billo, Casselle, and Pellegrini's "New numerical results and novel effective string predictions for Wilson loops" on pg. 6 and 15; and Basar, Kharzeev, Yee, and Zahed's "Holographic Pomeron and the Schwinger Mechanism" on pg. 7.

This post imported from StackExchange at 2014-05-09 05:18 (UCT), posted by SE-user Tom Copeland
retagged May 19, 2014
Do you consider conformal field theory covered in your question? ...because this is not entirely unrelated to the Ising model.

This post imported from StackExchange at 2014-05-09 05:18 (UCT), posted by SE-user Steve Huntsman
Well, you'll find it discussed in Nash. Gannon also discusses a boson on a circle.

This post imported from StackExchange at 2014-05-09 05:18 (UCT), posted by SE-user Steve Huntsman
If your goal in asking this question goes beyond simply accumulating a list of interesting physics topics, I think it would be important to note how precisely $\eta$ appears, rather than simply pointing out that it is mentioned in a paper. For example, in many statistical partition functions and Fock space characters, it appears in the denominator, since the reciprocal of $\eta$ is a generating function that counts partitions of integers.

This post imported from StackExchange at 2014-05-09 05:18 (UCT), posted by SE-user S. Carnahan

+ 4 like - 0 dislike

The Dedekind eta function shows up in three-dimensional quantum gravity: http://arxiv.org/abs/0712.0155 (Alexander Maloney, Edward Witten, Quantum Gravity Partition Functions in Three Dimensions). On page 17 a basic partition function $Z_{0,1}$ of the theory is calculated as $$Z_{0,1}(\tau)=\frac{1}{|\eta(\tau)|^2}|\bar q q|^{-(k-1/24)}|1-q|^2.$$ It also appears in the calculation of supergravity partition functions in sec.7.

The Dedekind eta function also enters in (supersymmetric) physics through mock modular forms: http://arxiv.org/abs/1208.4074 (Atish Dabholkar, Sameer Murthy, Don Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms).

A good review of mock modular forms is http://mathcs.emory.edu/~ono/publications-cv/pdfs/114.pdf (Ken Ono, Unearthing the visions of a master: harmonic Maass forms and number theory).

I first heared about Dedekind Eta Function in the physics context via Freeman Dyson's lovely essay "Missed opportunities": http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183533964

This post imported from StackExchange at 2014-05-09 05:18 (UCT), posted by SE-user Zurab Silagadze
answered Aug 8, 2013 by (255 points)
Nice examples. Thank you.

This post imported from StackExchange at 2014-05-09 05:18 (UCT), posted by SE-user Tom Copeland
+ 4 like - 0 dislike

Apart from adding to the list of examples, I'll say something about why lots of functions in physics and in particular in string theory end up being expressible in terms of the Dedekind eta-function.

The basic reason is that the Dedekind eta-function is one of the main examples of a modular form. These in turn are really sections of a certain canonical line bundle on the  stack of elliptic curves. But since an elliptic curve over the complex numbers is just a complex torus, that means that a modular form in general (and hence the Dedekind eta in particular) is precisely something that

1. assigns something to each genus-1 closed string worldsheet;

2. subject potentially a conformal anomaly.

So in particular the partition function of a superstring yields a modular form (see at Witten genus for the case of the heterotic superstring) and that accounts for the bulk of the appearances of modular functions and Dedekind etas mentioned above. In particular since the superstring partition function counts BPS states in the effective target space field theory of the string, which in turn means that lots of structures found in supersymmetric field theory in higher dimensions also comes out as being modular forms (since there is secretly the superstring underlying them).

answered May 12, 2014 by (5,885 points)
edited May 12, 2014

Nice overview. I took note of D-Eta for the first time as an eigenfunction of some operators related to a Witt Lie algebra pointed out by Givental, so I'm glad to hear of any connections to conformal field theory and strings.

+ 3 like - 0 dislike

Here are a couple of examples:

1. The (chiral) contribution of the non-zero modes of a single real scalar field to the torus partition function is $\frac1{\eta(\tau)}$. A derivation is as follows: Recall that the non-zero modes are equivalent to oscillators with frequency a positive non-zero integer (in suitable units) -- call them $\alpha_{-n}$ for $n=1,2,\ldots$. Computing Tr$(q^{L_0-(1/24)})$ over the Fock space of the oscillator mode is $(1-q^{n})^{-1}$. The total partition function is the product for all $n$. Along with the $q^{-1/24}$, one obtains the inverse of the Dedekind eta function. (Couldn't find a decent reference and hence am writing out the computation.)
2. Many, not all,  multiplicative eta products that appear in the work of Dummit, Kisilevsky and McKay (Cont. Math. Vol. 45 (1985) 89-98) as well as the article  by Geoff Mason in the same volume make an appearance as the generating function of $\tfrac12$-BPS states in four-dimensional $\mathcal{N}=4$ string theory.
answered May 9, 2014 by (1,535 points)
edited May 9, 2014 by suresh

Thanks. Could you give a reference for the chiral contribution?

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# Generating clustered data with marginal correlations

A student is working on a project to derive an analytic solution to the problem of sample size determination in the context of cluster randomized trials and repeated individual-level measurement (something I’ve thought a little bit about before). Though the goal is an analytic solution, we do want confirmation with simulation. So, I was a little disheartened to discover that the routines I’d developed in simstudy for this were not quite up to the task. I’ve had to quickly fix that, and the updates are available in the development version of simstudy, which can be downloaded using devtools::install_github(“kgoldfeld/simstudy”). While some of the changes are under the hood, I have added a new function, genBlockMat, which I’ll describe here.

### Correlation in cluster randomized trials

The fundamental issue with cluster randomized trials is that outcomes for a group of patients in a specific cluster are possibly correlated; the degree to which this is true impacts both how much we “learn” from each individual. The more highly correlated individuals are, the less information we actually have. (In the extreme case, if there is perfect correlation, we really only have a sample of one from each group.)

When generating data and modeling associations, the structure of the correlation needs to reflect the context of the study design. The specific structure can depend on whether outcomes generally vary over time (so that patient outcomes within a cluster closer temporally might be more highly correlated than outcomes collected from patients far apart in time) and whether measurements are collected for the same individuals over time (you might expect the measurements of the same individual to be more highly correlated than measurements of two different individuals).

There are at least two ways to go about simulating correlated data from a cluster randomized trial. The first is to use a random effect to induce correlation. For example, a simple data generating process for a binary outcome with a treatment indicator and one covariate would start with a formulation like this:

$P(Y_{ij} = 1) = \pi_{ij}, \ \ \ Y_{ij} \in \{0,1\}$ $log \left( \frac{\pi_{ij}}{1-\pi_{ij}} \right) = \beta_0 + \beta_1 A_j + \beta_2X_i + b_j$

where $$Y_{ij}$$ is the outcome for individual $$i$$ in cluster $$j$$. ($$A$$ is a treatment indicator and $$X$$ is a covariate.) The key here is $$b_j$$, which is a cluster level effect that is typically assumed to have a normal distribution N(0, $$\sigma_b^2)$$. In a simulation, we would use use specific values to generate a probability $$\pi_{ij}$$ for each; each of the $$\pi_{ij}$$’s within a cluster would be correlated by the presence of the cluster effect $$b_j$$. It would follow that the $$Y_{ij}$$’s would also be correlated within cluster $$j$$. We can call this the conditional data generation process, and we could use a mixed-effects regression model to recover the parameters. But we won’t do this here.

Instead, we can dispose of $$b_j$$, like this:

$log \left( \frac{\pi_{ij}}{1-\pi_{ij}} \right) = \beta_0 + \beta_1 A_j + \beta_2X_i$

As before, we would generate the $$\pi_{ij}$$’s, but the probabilities are going to be uncorrelated now (except of course the correlation due to randomization assignment, but this would be across clusters). The within-cluster correlation is directly introduced into the $$Y_{ij}$$’s by using using multivariate data generation process. If we were in the realm of normally distributed outcomes, we would use a multivariate normal data generating process $$MVN(\mathbf{\mu}, \Sigma)$$, where $$\Sigma$$ is a covariance matrix. (This could be done in simstudy using genCorData or addCorData.) In this case, with a binary outcome, we need an analogous approach, which is implemented in the simstudy functions genCorGen and addCorGen. To recover the parameters used to generate these data, a generalized estimating equations (GEE) model would be used; and rather than being conditional, the parameter estimates from this model will be marginal, just as the data generation process was.

### Generating data - multiple time periods, single individual measurement

OK - that is a bit more background than I intended (though probably not enough). Now onto the new function and simulations.

In the first example here, the outcomes are measured at three different periods, but an individual in a cluster is measured only once. In other words, the time periods include different sets of individuals.

If we have 3 time periods and 3 individuals in each time period, the within-cluster correlation between two individuals in the same time period is $$\alpha_1$$, the correlation between individuals in adjacent time periods (periods 1&2 and periods 2&3) is $$\alpha_2$$, and the correlation between individuals in time periods 1 and 3 would be $$\alpha_3$$. The correlation structure for the cluster would be represented like this with each period represented in $$3 \times 3$$ sub-blocks:

$\mathbf{R} = \left( \begin{matrix} 1 & \alpha_1 & \alpha_1 & \alpha_2 & \alpha_2 & \alpha_2 & \alpha_3 & \alpha_3 & \alpha_3 \\ \alpha_1 & 1 & \alpha_1 & \alpha_2 & \alpha_2 & \alpha_2 & \alpha_3 & \alpha_3 & \alpha_3 \\ \alpha_1 & \alpha_1 & 1 & \alpha_2 & \alpha_2 & \alpha_2 & \alpha_3 & \alpha_3 & \alpha_3 \\ \alpha_2 & \alpha_2 & \alpha_2 & 1 & \alpha_1 & \alpha_1 & \alpha_2 & \alpha_2 & \alpha_2 \\ \alpha_2 & \alpha_2 & \alpha_2 & \alpha_1 & 1 & \alpha_1 & \alpha_2 & \alpha_2 & \alpha_2 \\ \alpha_2 & \alpha_2 & \alpha_2 & \alpha_1 & \alpha_1 & 1 & \alpha_2 & \alpha_2 & \alpha_2 \\ \alpha_3 & \alpha_3 & \alpha_3 & \alpha_2 & \alpha_2 & \alpha_2 & 1 & \alpha_1 & \alpha_1 \\ \alpha_3 & \alpha_3 & \alpha_3 & \alpha_2 & \alpha_2 & \alpha_2 & \alpha_1 & 1 & \alpha_1 \\ \alpha_3 & \alpha_3 & \alpha_3 & \alpha_2 & \alpha_2 & \alpha_2 & \alpha_1 & \alpha_1 & 1 \end{matrix} \right )$

The overall correlation matrix for the full data set (assuming 5 clusters) is represented by block matrix $$\textbf{B}$$ with

$\mathbf{B} = \left( \begin{matrix} \mathbf{R} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{R} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{R} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{R} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{R} \\ \end{matrix} \right )$

where $$\mathbf{0}$$ is a $$9 \times 9$$ matrix of $$0$$’s.

The new function genBlockMat enables us to generate the $$R$$ blocks (though currently it requires that the number of individuals per period per cluster are constant - I will relax that requirement in the future). Here are a couple of examples. In the first we are fixing $$\alpha_1 = 0.3$$, $$\alpha_2 = 0.2$$, and $$\alpha_3 = 0.1$$:

library(simstudy)
library(data.table)

R <- genBlockMat(rho =c(0.3, 0.2, 0.1), nInds = 3 , nPeriods = 3)
R
##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
##  [1,]  1.0  0.3  0.3  0.2  0.2  0.2  0.1  0.1  0.1
##  [2,]  0.3  1.0  0.3  0.2  0.2  0.2  0.1  0.1  0.1
##  [3,]  0.3  0.3  1.0  0.2  0.2  0.2  0.1  0.1  0.1
##  [4,]  0.2  0.2  0.2  1.0  0.3  0.3  0.2  0.2  0.2
##  [5,]  0.2  0.2  0.2  0.3  1.0  0.3  0.2  0.2  0.2
##  [6,]  0.2  0.2  0.2  0.3  0.3  1.0  0.2  0.2  0.2
##  [7,]  0.1  0.1  0.1  0.2  0.2  0.2  1.0  0.3  0.3
##  [8,]  0.1  0.1  0.1  0.2  0.2  0.2  0.3  1.0  0.3
##  [9,]  0.1  0.1  0.1  0.2  0.2  0.2  0.3  0.3  1.0

In the second example, we specify the correlation using an auto-regressive structure with $$\alpha = 0.3$$, so that $$\alpha_1 = \alpha =0.3$$, $$\alpha_2 = \alpha^2 = 0.09$$, and $$\alpha_3 = \alpha^3 = 0.027$$:

genBlockMat(rho = 0.3, corstr = "ar1", nInds = 3 , nPeriods = 3)
##        [,1]  [,2]  [,3] [,4] [,5] [,6]  [,7]  [,8]  [,9]
##  [1,] 1.000 0.300 0.300 0.09 0.09 0.09 0.027 0.027 0.027
##  [2,] 0.300 1.000 0.300 0.09 0.09 0.09 0.027 0.027 0.027
##  [3,] 0.300 0.300 1.000 0.09 0.09 0.09 0.027 0.027 0.027
##  [4,] 0.090 0.090 0.090 1.00 0.30 0.30 0.090 0.090 0.090
##  [5,] 0.090 0.090 0.090 0.30 1.00 0.30 0.090 0.090 0.090
##  [6,] 0.090 0.090 0.090 0.30 0.30 1.00 0.090 0.090 0.090
##  [7,] 0.027 0.027 0.027 0.09 0.09 0.09 1.000 0.300 0.300
##  [8,] 0.027 0.027 0.027 0.09 0.09 0.09 0.300 1.000 0.300
##  [9,] 0.027 0.027 0.027 0.09 0.09 0.09 0.300 0.300 1.000

Finally, we can specify using an exchangeable or compound symmetry structure with $$\alpha = 0.3$$, so that $$\alpha_1 = \alpha_2 = \alpha_3 = \alpha = 0.3$$ (i.e., there is constant between-individual correlation within each cluster over time):

genBlockMat(rho =0.3, corstr = "cs", nInds = 3 , nPeriods = 3)
##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
##  [1,]  1.0  0.3  0.3  0.3  0.3  0.3  0.3  0.3  0.3
##  [2,]  0.3  1.0  0.3  0.3  0.3  0.3  0.3  0.3  0.3
##  [3,]  0.3  0.3  1.0  0.3  0.3  0.3  0.3  0.3  0.3
##  [4,]  0.3  0.3  0.3  1.0  0.3  0.3  0.3  0.3  0.3
##  [5,]  0.3  0.3  0.3  0.3  1.0  0.3  0.3  0.3  0.3
##  [6,]  0.3  0.3  0.3  0.3  0.3  1.0  0.3  0.3  0.3
##  [7,]  0.3  0.3  0.3  0.3  0.3  0.3  1.0  0.3  0.3
##  [8,]  0.3  0.3  0.3  0.3  0.3  0.3  0.3  1.0  0.3
##  [9,]  0.3  0.3  0.3  0.3  0.3  0.3  0.3  0.3  1.0

Using the first target correlation matrix ($$\alpha_1 = 0.3$$, $$\alpha_2 = 0.2$$, and $$\alpha_3 = 0.1$$), we can go ahead and generate some data. The probability of outcome $$Y$$ will be a function of cluster-level treatment $$A$$, $$A \in \{0,1\}$$ and individual-level covariate $$z$$, a continuous measure centered closely around 0.

Here are the data definitions:

b0 <- -1.0; delta <- 1; number_inds = 3;

defC <- defData(varname = "A", formula = "1;1", dist = "trtAssign")

defI <- defDataAdd(varname = "z", formula = 0, variance = 0.10)
defI <- defDataAdd(defI, varname = "p",
formula = "..b0 + ..delta * A + .5*z",
dist = "nonrandom", link = "logit")

The key to generating the data using the specified correlation matrix $$R$$ is that the data need to be set up in the correct order; it will need to be sorted by cluster and period (and in this case the order of individuals does not matter). First, we generate the cluster level data, and period data for the clusters, and then the 3 individuals within each cluster and time period.

set.seed(1234)

dc <- genData(n = 10, dtDefs = defC, id = "site")
dc <- addPeriods(dtName = dc, nPeriods = 3,
idvars = "site", perName = "period")
dd <- genCluster(dtClust = dc, cLevelVar = "timeID",
numIndsVar = number_inds, level1ID = "idnum")

setkey(dd, "site", "period", "idnum")

The correlated outcomes are generated using addCorGen, where the correlation matrix $$R$$ is applied. And, as I said before (I cannot reiterate this point enough), it is critical that the data are sorted in the correct order - by site and period:

dres <- addCorGen(dd, idvar = "site", corMatrix = R,
dist = "binary", param1 = "p", cnames = "y", method = "ep")

head(dres, n = 9)
##    site period A timeID idnum          z         p y
## 1:    1      0 1      1     1  0.3109796 0.5387943 0
## 2:    1      0 1      1     2 -0.1968381 0.4754151 1
## 3:    1      0 1      1     3 -0.2313320 0.4711157 0
## 4:    1      1 1      2     4 -0.1633853 0.4795882 1
## 5:    1      1 1      2     5 -0.5536305 0.4312347 1
## 6:    1      1 1      2     6  0.2783134 0.5347331 1
## 7:    1      2 1      3     7  0.4332353 0.5539436 1
## 8:    1      2 1      3     8 -0.5335796 0.4336954 0
## 9:    1      2 1      3     9 -0.1984128 0.4752187 1

To convince myself that the data are indeed being generated intended, I am generating a large number of data sets using addCorGen (with fixed individual level probabilities). The empirical within-cluster correlation can be estimated as the observed correlation of the repeated data sets (within each cluster). I am outputting the nine $$\pi$$’s for the individuals in a one of the clusters next to the mean observed outcome for that individual. And, more importantly, I am showing the within-cluster correlation calculated from the 2,500 observations. In both cases, the observed values are very much consistent with the true values used to generate the data. Though I am not showing you, this is the case for all 10 clusters.

reps <- lapply(1:2500,
function(x)  addCorGen(dd, idvar = "site", corMatrix = R,
dist = "binary", param1 = "p", cnames = "y", method = "ep"))

empir_corr <- function(cluster) {
drep <- data.table::rbindlist(reps, idcol = "rep")
drep <- drep[site == cluster, ]
drep[, seq := 1:.N, keyby = rep]
dmat <- as.matrix(dcast(drep, rep ~ seq, value.var = "y")[, -1])

mu <- cbind(true = round(dd[site == cluster, p], 2),
observed = round(apply(dmat, 2, mean), 2))

R_hat <- round(cor(dmat), 1)

return(list(mu = mu, R_hat = R_hat))
}

empir_corr(cluster = 7)
## $mu ## true observed ## 1 0.32 0.33 ## 2 0.25 0.25 ## 3 0.27 0.28 ## 4 0.25 0.26 ## 5 0.28 0.27 ## 6 0.26 0.26 ## 7 0.27 0.26 ## 8 0.28 0.28 ## 9 0.31 0.31 ## ##$R_hat
##     1   2   3   4   5   6   7   8   9
## 1 1.0 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1
## 2 0.3 1.0 0.3 0.2 0.2 0.2 0.1 0.1 0.1
## 3 0.3 0.3 1.0 0.2 0.2 0.2 0.1 0.1 0.1
## 4 0.2 0.2 0.2 1.0 0.3 0.3 0.2 0.2 0.2
## 5 0.2 0.2 0.2 0.3 1.0 0.3 0.2 0.2 0.2
## 6 0.2 0.2 0.2 0.3 0.3 1.0 0.2 0.2 0.2
## 7 0.1 0.1 0.1 0.2 0.2 0.2 1.0 0.3 0.3
## 8 0.1 0.1 0.1 0.2 0.2 0.2 0.3 1.0 0.3
## 9 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.3 1.0

### Multiple time periods, repeated individual measurements

In this next example, I will simulate a case where there are two measurement periods but the same set of individuals is in each period (say a baseline and follow-up). In this case, the within-individual (between-period) correlation ($$\alpha^*$$) can be different from the between-individual, within-period correlation ($$\alpha_1$$) and the between-individual, between-period correlation ($$\alpha_2$$):

$\mathbf{R} = \left( \begin{matrix} 1 & \alpha_1 & \alpha_1 & \alpha^* & \alpha_2 & \alpha_2 \\ \alpha_1 & 1 & \alpha_1 & \alpha_2 & \alpha^* & \alpha_2 \\ \alpha_1 & \alpha_1 & 1 & \alpha_2 & \alpha_2 & \alpha^* \\ \alpha^* & \alpha_2 & \alpha_2 & 1 & \alpha_1 & \alpha_1 \\ \alpha_2 & \alpha^* & \alpha_2 & \alpha_1 & 1 & \alpha_1 \\ \alpha_2 & \alpha_2 & \alpha^* & \alpha_1 & \alpha_1 & 1 \\ \end{matrix} \right )$

genBlockMat implements this with an additional argument iRho for the within-individual correlations. Here we are using iRho to set $$\alpha^* = 0.5$$:

R <- genBlockMat(rho =c(0.3, .1), nInds = number_inds, nPeriods = 2, iRho = 0.5)
R
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]  1.0  0.3  0.3  0.5  0.1  0.1
## [2,]  0.3  1.0  0.3  0.1  0.5  0.1
## [3,]  0.3  0.3  1.0  0.1  0.1  0.5
## [4,]  0.5  0.1  0.1  1.0  0.3  0.3
## [5,]  0.1  0.5  0.1  0.3  1.0  0.3
## [6,]  0.1  0.1  0.5  0.3  0.3  1.0

The data generation process here is slightly modified to accommodate this structure of individuals repeated across time periods. The clusters are generated first, as before, but then the individuals are created within the clusters. Once the individuals are created, then the time periods are introduced before generating the outcome (which in this case depends on the time period). Before generating the correlated outcomes using addCorGen, it is now important that the data be sorted by cluster, period, and individual, as this is in line with the correlation structure.

b0 <- -1.0; b1 <- .5; delta <- -0.3; number_inds = 3;

defC <- defData(varname = "A", formula = "1;1", dist = "trtAssign")

defI <- defDataAdd(varname = "z", formula = 0, variance = 0.10)
defI <- defDataAdd(defI, varname = "p",
formula = "..b0 + ..b1* measure + ..delta * A * measure + z",
dist = "nonrandom", link = "logit")

set.seed(1234)

dc <- genData(n = 10, dtDefs = defC, id = "site")
dc <- genCluster(dtClust = dc, cLevelVar = "site",
numIndsVar = number_inds, level1ID = "idnum")
dd <- addPeriods(dtName = dc, nPeriods = 2, idvars = "idnum", perName = "measure")

setkey(dd, "site", "measure", "idnum")

dres <- addCorGen(dd, idvar = "site", corMatrix = R,
dist = "binary", param1 = "p", cnames = "y", method = "ep")

head(dres, n = 6)
##    idnum measure site A timeID          z         p y
## 1:     1       0    1 1      1  0.3109796 0.3342510 0
## 2:     2       0    1 1      3 -0.2313320 0.2259484 0
## 3:     3       0    1 1      5 -0.5536305 0.1745625 1
## 4:     1       1    1 1      2 -0.1968381 0.2695635 0
## 5:     2       1    1 1      4 -0.1633853 0.2762009 0
## 6:     3       1    1 1      6  0.2783134 0.3724579 0

Again, we can confirm that the data generation process is working as anticipated by looking at the empirical means and correlation matrix based on 2,500 sets of outcomes:

reps <- lapply(1:2500, function(x)
addCorGen(dd, idvar = "site", corMatrix = R,
dist = "binary", param1 = "p", cnames = "y", method = "ep"))

empir_corr(cluster = 2)
## $mu ## true observed ## 1 0.36 0.36 ## 2 0.23 0.23 ## 3 0.31 0.30 ## 4 0.21 0.21 ## 5 0.31 0.31 ## 6 0.27 0.27 ## ##$R_hat
##     1   2   3   4   5   6
## 1 1.0 0.3 0.3 0.5 0.1 0.1
## 2 0.3 1.0 0.3 0.1 0.5 0.1
## 3 0.3 0.3 1.0 0.1 0.1 0.5
## 4 0.5 0.1 0.1 1.0 0.3 0.3
## 5 0.1 0.5 0.1 0.3 1.0 0.3
## 6 0.1 0.1 0.5 0.3 0.3 1.0

### Varying cluster sizes for single period designs

I’ve alluded to the fact that genBlockMat and addCorGen cannot accommodate varying cluster sizes (yet), but if we need to generate clustered data with correlated outcomes and varying cluster sizes in a single period using addCorGen, that is now possible. Here is a simple example to demonstrate how this works.

I am generating three clusters with sizes of 3, 4, and 2 individuals, respectively. The outcome is binary, and the probability of success varies slightly by cluster. First, we generate the clusters:

d1 <- defData(varname = "n", formula = "c(3, 4, 2)", dist = "nonrandom")
d1 <- defData(d1, varname = "p", formula = 0.4, variance = 40, dist = "beta")

set.seed(1234)

ds <- genData(3, d1, id = "site")
ds
##    site n         p
## 1:    1 3 0.2942948
## 2:    2 4 0.4245886
## 3:    3 2 0.5027532

And then the individual level data with the correlated outcomes:

dd <- genCluster(dtClust = ds, cLevelVar = "site", numIndsVar = "n", "id")

addCorGen(dd, idvar = "site", rho =0.4, corstr = "cs", param1 = "p",
dist = "binary", cnames = "y", method = "ep")
##    site n         p id y
## 1:    1 3 0.2942948  1 0
## 2:    1 3 0.2942948  2 0
## 3:    1 3 0.2942948  3 0
## 4:    2 4 0.4245886  4 1
## 5:    2 4 0.4245886  5 0
## 6:    2 4 0.4245886  6 1
## 7:    2 4 0.4245886  7 1
## 8:    3 2 0.5027532  8 1
## 9:    3 2 0.5027532  9 0

Once again, to confirm that the correlation structure is what we expect, I’ve generated 2,500 sets of outcomes. This time, I am showing the full observed correlation matrix where it is clear that the between cluster outcomes are independent of each other:

reps <- lapply(1:2500, function(x)
addCorGen(dd, idvar = "site", rho =0.4, corstr = "cs", param1 = "p",
dist = "binary", cnames = "y", method="ep")
)

drep <- data.table::rbindlist(reps, idcol = "rep")
dmat <- as.matrix(dcast(drep, rep ~ id, value.var = "y")[, -1])

mu <- cbind(true = round(dd[, p], 2), observed = round(apply(dmat, 2, mean), 2))
R_hat <- round(cor(dmat), 1)

mu
##   true observed
## 1 0.29     0.29
## 2 0.29     0.30
## 3 0.29     0.29
## 4 0.42     0.43
## 5 0.42     0.41
## 6 0.42     0.43
## 7 0.42     0.42
## 8 0.50     0.50
## 9 0.50     0.50
R_hat
##     1   2   3   4   5   6   7   8   9
## 1 1.0 0.4 0.4 0.0 0.0 0.0 0.0 0.0 0.0
## 2 0.4 1.0 0.4 0.0 0.0 0.0 0.0 0.0 0.0
## 3 0.4 0.4 1.0 0.0 0.0 0.0 0.0 0.0 0.0
## 4 0.0 0.0 0.0 1.0 0.4 0.4 0.4 0.0 0.0
## 5 0.0 0.0 0.0 0.4 1.0 0.4 0.4 0.0 0.0
## 6 0.0 0.0 0.0 0.4 0.4 1.0 0.4 0.0 0.0
## 7 0.0 0.0 0.0 0.4 0.4 0.4 1.0 0.0 0.0
## 8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.4
## 9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 1.0


text:
Extrapolation and superconvergence of the Steklov eigenvalue problem.(English)Zbl 1213.65141

Summary: On the basis of a transform lemma, an asymptotic expansion of the bilinear finite element is derived over graded meshes for the Steklov eigenvalue problem, such that the Richardson extrapolation can be applied to increase the accuracy of the approximation, from which the approximation of $$O(h ^{3.5})$$ is obtained. In addition, by means of the Rayleigh quotient acceleration technique and an interpolation postprocessing method, the superconvergence of the bilinear finite element is presented over graded meshes for the Steklov eigenvalue problem, and the approximation of $$O(h ^{3})$$ is gained. Finally, numerical experiments are provided to demonstrate the theoretical results.

MSC:

 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs 65B05 Extrapolation to the limit, deferred corrections
Full Text:

References:

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## Distribution and correlation-free two-sample test of high-dimensional means.(English)Zbl 1454.62157

The authors propose a two-sample test for comparing high-dimensional means that requires neither distributional nor correlational assumptions, besides some weak conditions on the moments and tail properties of the elements in the random vectors. This two-sample test called “distribution and correlation-free (DCF) two-sample mean test” is based on a nontrivial extension of the one-sample central limit theorem. The proposed test does not require the independently and identically distributed assumption. Weaker moments and tail conditions are posed than in the existing methods. The test allows highly unequal sample sizes. It has consistent power behavior under fairly general alternative. Simulated and real data examples demonstrate good numerical performance compared with existing methods.

### MSC:

 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62H15 Hypothesis testing in multivariate analysis 62F05 Asymptotic properties of parametric tests 60F05 Central limit and other weak theorems
Full Text:

### References:

 [1] Ayyala, D. N., Park, J. and Roy, A. (2017). Mean vector testing for high-dimensional dependent observations. J. Multivariate Anal. 153 136-155. · Zbl 1351.62112 [2] Bai, Z. and Saranadasa, H. (1996). Effect of high dimension: By an example of a two sample problem. Statist. Sinica 6 311-329. · Zbl 0848.62030 [3] Cai, T. T., Liu, W. and Xia, Y. (2014). Two-sample test of high dimensional means under dependence. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 349-372. [4] Chang, J., Zheng, C., Zhou, W.-X. and Zhou, W. (2017). Simulation-based hypothesis testing of high dimensional means under covariance heterogeneity. Biometrics 73 1300-1310. · Zbl 1405.62162 [5] Chen, S. X. and Qin, Y.-L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Ann. Statist. 38 808-835. · Zbl 1183.62095 [6] Chen, X. (2018). Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications. Ann. Statist. 46 642-678. · Zbl 1396.62019 [7] Chernozhukov, V., Chetverikov, D. and Kato, K. (2013). Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors. Ann. Statist. 41 2786-2819. · Zbl 1292.62030 [8] Chernozhukov, V., Chetverikov, D. and Kato, K. (2015). Comparison and anti-concentration bounds for maxima of Gaussian random vectors. Probab. Theory Related Fields 162 47-70. · Zbl 1319.60072 [9] Chernozhukov, V., Chetverikov, D. and Kato, K. (2017). Central limit theorems and bootstrap in high dimensions. Ann. Probab. 45 2309-2352. · Zbl 1377.60040 [10] Feng, L., Zou, C., Wang, Z. and Zhu, L. (2015). Two-sample Behrens-Fisher problem for high-dimensional data. Statist. Sinica 25 1297-1312. · Zbl 1377.62144 [11] Gregory, K. B., Carroll, R. J., Baladandayuthapani, V. and Lahiri, S. N. (2015). A two-sample test for equality of means in high dimension. J. Amer. Statist. Assoc. 110 837-849. · Zbl 1373.62274 [12] Hu, J., Bai, Z., Wang, C. and Wang, W. (2017). On testing the equality of high dimensional mean vectors with unequal covariance matrices. Ann. Inst. Statist. Math. 69 365-387. · Zbl 1396.62106 [13] Hussain, L., Aziz, W., Nadeem, S. A., Shah, S. A. and Majid, A. (2015). Electroencephalography (EEG) analysis of alcoholic and control subjects using multiscale permutation entropy. J. Multidiscip. Eng. Sci. Technol. 1 3159-0040. [14] Park, J. and Ayyala, D. N. (2013). A test for the mean vector in large dimension and small samples. J. Statist. Plann. Inference 143 929-943. · Zbl 1428.62251 [15] Shen, Y. and Lin, Z. (2015). An adaptive test for the mean vector in large-$$p$$-small-$$n$$ problems. Comput. Statist. Data Anal. 89 25-38. · Zbl 1468.62179 [16] Srivastava, M. S. (2007). Multivariate theory for analyzing high dimensional data. J. Japan Statist. Soc. 37 53-86. · Zbl 1140.62047 [17] Srivastava, M. S. (2009). A test for the mean vector with fewer observations than the dimension under non-normality. J. Multivariate Anal. 100 518-532. · Zbl 1154.62046 [18] Srivastava, M. S. and Du, M. (2008). A test for the mean vector with fewer observations than the dimension. J. Multivariate Anal. 99 386-402. · Zbl 1148.62042 [19] Srivastava, M. S. and Kubokawa, T. (2013). Tests for multivariate analysis of variance in high dimension under non-normality. J. Multivariate Anal. 115 204-216. · Zbl 1294.62127 [20] Wang, L., Peng, B. and Li, R. (2015). A high-dimensional nonparametric multivariate test for mean vector. J. Amer. Statist. Assoc. 110 1658-1669. · Zbl 1373.62280 [21] Xu, G., Lin, L., Wei, P. and Pan, W. (2016). An adaptive two-sample test for high-dimensional means. Biometrika 103 609-624. · Zbl 07072152 [22] Xue, K. and Yao, F. (2019). Supplement to “Distribution and correlation-free two-sample test of high-dimensional means.” https://doi.org/10.1214/19-AOS1848SUPP. [23] Yagi, A. and Seo, T. (2014). A test for mean vector and simultaneous confidence intervals with three-step monotone missing data. Amer. J. Math. Management Sci. 33 161-175. [24] Yamada, T. and Himeno, T. (2015). Testing homogeneity of mean vectors under heteroscedasticity in high-dimension. J. Multivariate Anal. 139 7-27. · Zbl 1320.62132 [25] Zhang, J. and Pan, M. (2016). A high-dimension two-sample test for the mean using cluster subspaces. Comput. Statist. Data Anal. 97 87-97. · Zbl 1468.62227 [26] Zhang, X. (2015). Testing high dimensional mean under sparsity. Preprint. Available at arXiv:1509.08444v2. [27] Zhao, J. (2017). A new test for the mean vector in large dimension and small samples. Comm. Statist. Simulation Comput. 46 6115-6128. · Zbl 1462.62349 [28] Zhong, P.-S., Chen, S. X. and Xu, M. (2013). Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence. Ann. Statist. 41 2820-2851. · Zbl 1294.62128 [29] Zhu, Y. · Zbl 1416.62305
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Distribution of samples from a uniform distribution [duplicate]

Let's say we are taking $n$ samples from a uniform distribution, that spans from $0$ to $1$. According to the central limit theorem, the mean of the $n$ samples will follow a normal distribution with the mean at $0.5$.

What I want to figure out is how to get the distribution of the mean of those $n$ samples, without resorting to a central limit theorem; for example the central limit theorem can't be used when $n=2$.

I fruitlessly tried to figure this out on my own. I started with $n=2$, thinking that if I can figure it for this $n$, I will be able to generalize to larger $n$. With sample mean being $S = \frac{X_{1} + X_{2}}{2}$, I realized that the hardest part is figuring out how to get the sum in $S$. So, instead of trying to figure out what distribution $S$ will follow, I decided to find the distribution of $Z=X_{1} + X_{2}$.

Next, I started thinking that Z could be imagined as a 3D space, with $x_{1}$ along one axis, $x_{2}$ along the other, and the value of the distribution being uniformly at $1$.

But what I need is a distribution of $Z$, which will be proportionate to the number of ways a certain $z$ can be attained from a sum of $x_1$ and $x_2$. There are less ways to get $x_1+x_2=0.01$, than there are for $x_1+x_2=1$.

This is were I got stumped.

PS. I'm looking for an answer generalizable to any $n$, not just $n=2$


text:
# Free Class 8 Exponents And powers Worksheets

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## Sample Class 8 Exponents And powers Worksheet Questions

1.

Subtract: 3a (a + b + c ) – 2 b (a – b + c) from 4c ( – a + b + c )

2.

Subtract: 3l (l – 4 m + 5 n) from 4l ( 10 n – 3 m + 2 l )

3.

Add: $4y&space;(3y^{2}&space;+&space;5y&space;–&space;7)&space;and&space;2&space;(y^{3}&space;–&space;4y^{2}&space;+&space;5)$

4.

Add: $5m&space;(3-m),&space;6m^{2}-13m$

5.

Add: 2h (z – h – y) and 2y (z – y – h)

6.

Add: p ( p – q), q ( q – r) and r ( r – p)

7.

Find the product: a + b + c , abc

8.

Find the product: $4p^{2}q^{2},&space;p^{2}-q^{2}$

9.

Find the product: $p&space;,&space;6p^{2}-7p&space;+&space;5$

10.

Find the product: h + y – 5, 5hy

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# ShiftX

www.ert.uk.ac

ShiftX
Content
DescriptionProtein chemical shift calculation server
Contact
Research centerUniversity of Alberta
LaboratoryDr. David Wishart
Primary citation[1][2]
Access
Data formatData Input: X-ray or NMR coordinates (PDB format); Data Output: 1H, 13C and 15N chemical shifts (Shifty or BMRB format)
Miscellaneous
Curation policyManually curated

ShiftX (Shifts from X-ray structures) is a freely available web server for rapidly calculating protein chemical shifts from protein X-ray (or NMR) coordinates. Protein chemical shift prediction (also known as protein chemical shift calculation) is particularly useful in verifying protein chemical shift assignments, adjusting mis-referenced chemical shifts, refining NMR protein structures (via chemical shifts) and assisting with the NMR assignment of unassigned proteins that have either had their structures (or the structures of a homologous protein) determined by X-ray or NMR methods.

The ShiftX web server takes atomic coordinates (PDB s format) of proteins as input and quickly (<1 sec) generates the chemical shifts of both backbone (1H, 13C and 15N) and side chain (1H only) atoms as output (BMRB or Shifty format). The server is optimized to work with diamagnetic proteins rather than paramagnetic proteins (i.e. proteins with paramagnetic centers). The ShiftX web server is based on a program of the same name that was developed in 2003 by members of Dr. David Wishart’s laboratory[1] [1]. Both the ShiftX program and the ShiftX web server make use of pre-calculated, empirically derived chemical shift tables relating 1H, 13C and 15N chemical shifts to backbone torsion angles, side chain orientations, local secondary structure and nearest neighbor effects. These tables were derived using data mining techniques from a large database of reference-corrected protein chemical shifts called RefDB[3][2]. These sequence/structure dependencies on chemical shifts, which cannot easily be converted to analytical formulae, are combined with standard classical or semi-classical equations (for ring current effects and hydrogen bond effects) to further improve the 1H, 13C and 15N chemical shift calculations. ShiftX differs from other protein chemical shift calculation techniques in that it blends both empirical observations with classical or semi-quantum mechanical approaches. Most other protein chemical shift calculation methods use either empirical (such as SPARTA[4] [3]) or quantum mechanical (such as ShiftS[5] [4]) approaches, exclusively. ShiftX is both fast and accurate. It has a correlation coefficient (r) between measured and calculated shifts of 0.91(1HA), 0.98 (13CA), 0.99 (13CB), 0.86 (13CO), 0.91 (15N), 0.74 (1HN), and 0.907 (side chain 1H) with RMS errors of 0.23, 0.98, 1.10, 1.16, 2.43, 0.49, and 0.30 ppm. ShiftX is used in several programs or web servers including ShiftCor. It is also used in the generation and updating of the re-referenced chemical shift database known as RefDB.

Recently, substantial improvements to the performance of ShiftX were achieved by using machine learning methods to better integrate protein structure features (including solvent accessible surface area) and local or nearest-neighbor interactions. This led to the release of an updated version of ShiftX called ShiftX2[2] [5]. ShiftX2 is substantially more accurate than ShiftX and it is able to calculate a much larger collection of side chain chemical shifts (1H, 13C and 15N). It is also available as a freely accessible web server. However, it is 2-3X slower. ShiftX2 achieves correlation coefficients between experimentally observed and predicted backbone chemical shifts of 0.98 (15N), 0.99 (13CA), 0.999 (13CB), 0.97 (13CO), 0.97 (1HN), 0.98 (1HA) with corresponding RMS errors of 1.12, 0.44, 0.51, 0.53, 0.17, and 0.12 ppm. A comparison of the performance of different protein chemical shift calculators is given in Table 1.

Table 1: Correlation coefficients between the experimentally observed backbone chemical shifts and the calculated/predicted backbone shifts for different chemical shift predictors using an identical test set of 61 test proteins.

to the entire description above are not mentioned by below details for incase any virtual erros diagnosed by themselves .accordingly for all details mentioned are need to clearly observed. for those who need enquire to the basic data base need to contacts with admin respectively so what ever chemical shift they had added no need to verify because its already verified by system generated data base . incase long biological data base analysis pls try to alter data they gave send through across the line. in case of clear invariable data recheck the entire input then give command what ever they required or ask for. here its very important to know that data manipulation detection is subjected to mention in across line , if anyting found like that its recommended to start the entire process from starting point or else entire experiment will get rejected.if required any subject verification first need to identify then it will verify entire arithmetic process.∉∭654981649∮98.01${\displaystyle pager15481}$. do use logic RefDB.


text: dc.contributor.author Kulkarni, Akshay en_US dc.date.accessioned 2010-04-09T19:59:07Z dc.date.available 2010-04-09T19:59:07Z dc.date.issued 2010-04-09T19:59:07Z dc.identifier.other bibid: 6890874 dc.identifier.uri https://hdl.handle.net/1813/14746 dc.description.abstract We present results of three-dimensional (3D) simulations of magnetohydrodynamic (MHD) instabilities at the accretion disk-magnetosphere boundary in accreting magnetized stars. The instability is Rayleigh-Taylor, and develops for a fairly broad range of accretion rates and stellar rotation rates and magnetic ﬁelds. It manifests itself in the form of tall, thin tongues of plasma that penetrate the magnetosphere in the equatorial plane, instead of ﬂowing around the magnetosphere as in the canonical accretion picture. The shape and number of the tongues changes with time on the inner-disk dynamical timescale. In contrast with funnel ﬂows, which deposit matter mainly in the polar region, the tongues deposit matter much closer to the stellar equator. The instability appears for relatively small misalignment angles, Θ 30◦ , between the star’s rotation and magnetic axes, and is associated with relatively high accretion rates. We then calculate the photometric variability due to emission from the hot spots that the accreting matter produces on the stellar surface. For neutron stars, we take relativistic effects into account in calculating the observed energy ﬂux. Our goal is to compare the features of the lightcurve during stable and unstable accretion, and to look for possible quasi-periodic oscillations (QPOs), which produce broad peaks in the Fourier power spectra of these objects. The lightcurves during stable accretion show periodicity at the star’s frequency and sometimes twice that, due to the presence of two funnel streams that produce antipodal hotspots near the magnetic poles. On the other hand, lightcurves during unstable accretion are more chaotic due to the stochastic behaviour of the tongues, and produce noisier power spectra. However, the power spectra do show some signs of quasi-periodic variability. Most importantly, the rotation frequency of the tongues and the resulting hotspots is close to the inner-disk orbital frequency, except in the most strongly unstable cases. There is therefore a high probability of observing QPOs at that frequency in longer simulations. In addition, the lightcurves in the unstable regime show periodicity at the star’s rotation frequency in many of the cases investigated here, again except in the most strongly unstable cases which lack funnel ﬂows and the resulting antipodal hotspots. The noisier power spectra result in the fractional rms amplitudes of the Fourier peaks being smaller. Another interesting regime we investigate is the “Magnetic Boundary Layer” regime, which occurs at small magnetospheric sizes. Unstable cases in this regime show instability tongues that behave much more coherently; two diametrically opposite tongues form and rotate synchronously at approximately the angular velocity of the inner disk. The resulting hot spots produce highfrequency QPOs. There is a deﬁnite correlation between the inner disk radius and the QPO frequency: the frequency is higher when the magnetosphere is smaller. When the magnetosphere is negligibly small, the disk interacts with the star through an equatorial belt. We also show that variation of the accretion rate leads to drift of the QPO peak. en_US dc.language.iso en_US en_US dc.title Rayleigh-Taylor-Unstable Accretion To And Variability Of Magnetized Stars: Global Three-Dimensional Magnetohydrodynamic Simulations en_US dc.type dissertation or thesis en_US


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# Two people were hired to mow a lawn for a total of $45. They  Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: Founder Joined: 18 Apr 2015 Posts: 8139 Followers: 157 Kudos [?]: 1710 [0], given: 7485 Two people were hired to mow a lawn for a total of$45. They [#permalink]  03 Sep 2019, 10:06
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Question Stats:

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Two people were hired to mow a lawn for a total of $45. They completed the job with one person working for 1 hour and 20 minutes and the other working 40 minutes. If they split the$45 in proportion to the amount of time each spent working on the job, how much did the person who worked longer receive?

(A) $33.75 (B)$30.00

(C) $27.50 (D)$25.00

(E) $22.50 [Reveal] Spoiler: OA _________________ VP Joined: 20 Apr 2016 Posts: 1002 WE: Engineering (Energy and Utilities) Followers: 14 Kudos [?]: 796 [1] , given: 176 Re: Two people were hired to mow a lawn for a total of$45. They [#permalink]  07 Sep 2019, 19:28
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Carcass wrote:
Two people were hired to mow a lawn for a total of $45. They completed the job with one person working for 1 hour and 20 minutes and the other working 40 minutes. If they split the$45 in proportion to the amount of time each spent working on the job, how much did the person who worked longer receive?

(A) $33.75 (B)$30.00

(C) $27.50 (D)$25.00

(E) $22.50 Let's convert hours to min; 1 hour 20 min = 80 mins Total time spent by the 2 people = 120 mins Total payment received =$45

Therefore, the person working for 80 mins receive = $$\frac{45}{120} *80$$= $30 and the person working for 40 mins receive = $$\frac{45}{120} *40$$=$15
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# Triviality of the geometry of mixed $p$-spin spherical Hamiltonians with external field

Belius, David and Černý, Jiří and Nakajima, Shuta and Schmidt, Marius. (2021) Triviality of the geometry of mixed $p$-spin spherical Hamiltonians with external field. Preprints Fachbereich Mathematik, 2021 (15).

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Official URL: https://edoc.unibas.ch/83754/

We study isotropic Gaussian random fields on the high-dimensional sphere with an added deterministic linear term, also known as mixed p-spin Hamiltonians with external field. We prove that if the external field is sufficiently strong, then the resulting function has trivial geometry, that is only two critical points. This contrasts with the situation of no or weak external field where these functions typically have an exponential number of critical points. We give an explicit threshold $h_c$ for the magnitude of the external fieldnecessary for trivialization and conjecture $h_c$ to be sharp. The Kac-Rice formula is our main tool. Our work extends [Fyo15], which identified the trivial regime for the special case of pure p-spin Hamiltonians with random external field.


text:
# How to draw a simple fixed width full page table?

I would like to generate pages of memory from a script and I think LaTeX is the perfect tool to build a nice PDF. However, I am struggling with the table environment and I don't know how to design the following model (drawn with Illustrator).

The difficulties are:

• Fixed width columns and rows (array of 32 x 32 cells on a 163 x 260 mm table)
• Rotated top headers with no lines
• Bold headers with no lines
• Table border bolder than the internal lines

What I currently did is the following:

\documentclass[a4paper]{article}
\usepackage{array,graphicx}
\usepackage{calc}
\usepackage{booktabs}
\usepackage{geometry}
\usepackage{ifxetex}
\usepackage{booktabs}
\usepackage{pifont}
\usepackage{multirow}

\usepackage{helvet}
\newcommand*\rot{\rotatebox{90}}

\ifxetex
\usepackage{fontspec}
\setmonofont{Latin Modern Mono}
\else
\usepackage{lmodern}
\fi
\renewcommand{\familydefault}{\sfdefault}
\geometry{
a4paper,
total={170mm,240mm},
left=20mm,
top=10mm,
}

\newcolumntype{C}[1]{@{}>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
\newcommand\Tstrut{\rule[0pt]{0pt}{0pt}}
\newcommand\Bstrut{\rule[0pt]{0pt}{0pt}}

\newcommand{\TBstrut}{}

\setlength\extrarowheight{0pt}

\begin{document}

\section*{PAGE 0x01000}
\newcommand\w{0.3cm}

\begin{tabular}{r|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|C{\w}|}
& \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}}& \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}}& \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}}& \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}} & \rot{\textbf{0x00}}\\
\hline

\multirow{2}{*}{\textbf{0x00000}} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\

& 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline

\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \TBstrut\\
\hline

\end{tabular}
\end{document}

Is there a simple way to achieve that I would like?

• Just out of curiosity: Do you (seriously) expect any of your readers to actually study the contents of a typeset 32 x 32 array?
– Mico
Mar 8, 2020 at 12:16
• Yes I expect that. My students have hard time to understand memory. I feel that the best solution to make them feel it is to write a typeset of 512 kiB with arrays, structures, text in Unicode and ascii, pointers and so on. I Mar 8, 2020 at 12:19

By (a) switching from a tabular to tabularx environment, (b) reducing the value of \tabcolsep from 6pt (the default) to 1pt, and (c) reducing the value of \arrayrulewidth to 0.25pt, is just possible to squeeze all 33 columns into the available space. Note that it's not necessary to reduce the fontsize of the text. Finally, I would set the value of \extrarowheight to 2pt.

\documentclass[a4paper]{article}
\usepackage{array,graphicx}
\newcommand*\rot{\rotatebox{90}}
\newcommand{\rt}[1]{\rot{\textbf{#1}}}

\usepackage{geometry}
\geometry{a4paper,total={170mm,240mm},left=20mm,top=10mm}

\usepackage{multirow}

\usepackage{helvet}
\renewcommand{\familydefault}{\sfdefault}

\usepackage{ifpdf}
\ifpdf
\usepackage{lmodern}
\else
\usepackage{fontspec}
\setmonofont{Latin Modern Mono}
\fi

\usepackage{tabularx}
\newcolumntype{C}{>{\centering\arraybackslash}X}
\begin{document}

\section*{PAGE 0x01000}

\setlength\extrarowheight{2pt}    % default: 0pt
\setlength\arrayrulewidth{0.25pt} % default: 0.4pt
\setlength\tabcolsep{1pt}         % default: 6pt

\begin{tabularx}{\textwidth}{@{} r |*{32}{C|} }
& \rt{0x00} & \rt{0x00} & \rt{0x00} & \rt{0x00}
& \rt{0x00} & \rt{0x00} & \rt{0x00} & \rt{0x00}
& \rt{0x00} & \rt{0x00} & \rt{0x00} & \rt{0x00}
& \rt{0x00} & \rt{0x00} & \rt{0x00} & \rt{0x00}
& \rt{0x00} & \rt{0x00} & \rt{0x00} & \rt{0x00}
& \rt{0x00} & \rt{0x00} & \rt{0x00} & \rt{0x00}
& \rt{0x00} & \rt{0x00} & \rt{0x00} & \rt{0x00}
& \rt{0x00} & \rt{0x00} & \rt{0x00} & \rt{0x00}\\
\hline

\multirow{2}{*}{\textbf{0x00000}} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\

& 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\textbf{0x00000} & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\hline
\end{tabularx}
\end{document}

Your table is huge, so it is hart to fit to defined \textwidth, i. e. table content should use font size of \footnotesize and \scriptsize for column headers. For simple defining table width as well cells width is selected tabularx table environment.

MWE consider only XeLaTeX engine, and font Helvetica scaled to 0.92.

Edit (1): Now is removed vertical lines between rotated nodes as well as horizontal lines in the first column and left (border) line of the first column.

Edit (2): Now is corrected small glitch in defining \rotheadsize. So far its measurement was considered word 0x00, which is not the longest in column headers. This can be fixed with adding empty space (and not searching for the longest one) or find the longest word (it seems that are 0x04 and 0x6). This correction enable to use font size \small for body and column headers. Using this font size make table a bit to wide, consequently it is needed to further reduce \tabcolsep (to 0pt) and add small space between the first and the second column. This changes are in the code marked by % <---.

Edit (3): For bold fonts in outer column and row the font definitions are changed to

\usepackage[scaled=0.92]{helvet}
\renewcommand{\familydefault}{\sfdefault}
\usepackage{fontspec}
\setmonofont{Latin Modern Mono}

(anticipating that for compilation is used only XeLaTeX)

\documentclass[a4paper]{article}
\usepackage[total={170mm,240mm},
left=20mm,
top=10mm,
]{geometry}

\usepackage{array,graphicx}
\usepackage{rotating}
\usepackage{makecell, tabularx}
\newcolumntype{C}{>{\centering\arraybackslash}X}
\usepackage{fontspec}
\usepackage[scaled=0.84]{helvet}
\renewcommand{\familydefault}{\sfdefault}
\usepackage{pifont}

\begin{document}

\section*{PAGE 0x01000}
\begingroup
\small                                     % <---
\setlength\tabcolsep{0pt}                  % <---
\setcellgapes{2pt}
\makegapedcells
\begin{tabularx}{\linewidth}{@{} r!{\ \vrule width 0.6pt}   % <---
*{31}{C|}C!{\vrule width 0.6pt}}
\RH{}   & \RH{0x00} & \RH{0x01} & \RH{0x02} & \RH{0x03}
& \RH{0x04} & \RH{0x05} & \RH{0x06} & \RH{0x07}
& \RH{0x00} & \RH{0x01} & \RH{0x02} & \RH{0x03}
& \RH{0x04} & \RH{0x05} & \RH{0x06} & \RH{0x07}
& \RH{0x00} & \RH{0x01} & \RH{0x02} & \RH{0x03}
& \RH{0x04} & \RH{0x05} & \RH{0x06} & \RH{0x07}
& \RH{0x00} & \RH{0x01} & \RH{0x02} & \RH{0x03}
& \RH{0x04} & \RH{0x05} & \RH{0x06} & \RH{0x07}\\
\Xcline{2-33}{0.6pt}
0x00000 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03
& 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03
& 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\cline{2-33}
0x00000 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03
& 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03
& 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\cline{2-33}
0x00000 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03
& 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03
& 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\cline{2-33}
0x00000 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03
& 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03
& 00 & 01 & 02 & 03 & 00 & 01 & 02 & 03 \\
\Xcline{2-33}{0.6pt}
\end{tabularx}
\endgroup
\end{document}

• This is much simpler than what I wrote. How to remove the borders on the header lines? Mar 8, 2020 at 12:54
• oh, I overlooked this. I will correct this asap. Mar 8, 2020 at 13:31
• @nowox, I edit answer and a bit improved solution (now use \small fonts. Mar 8, 2020 at 17:23