1922760350154212639070
メインコンテンツ
結果:
How can we use roughness in an effective context to identify large primes? I can quickly think of quite a few examples where we might do so. Again, remember I will be looking for primes with not just hundreds of decimal digits, or even only a few thousand digits. The eventual target is higher than that. Forget about targets for now though, as this is a journey, and what matters in this journey is what we may learn along the way.
I think the most obvious way to employ roughness is in a search for twin primes. Though not yet proven, the twin prime conjecture:
If it is true, it tells us there are infinitely many twin prime pairs. A twin prime pair is two integers with a separation of 2, such that both of them are prime. We can find quite a few of them at first, as we have {3,5}, {5,7}, {11,13}, etc. But there is only ONE pair of integers with a spacing of 1, such that both of them are prime. That is the pair {2,3}. And since primes are less and less common as we go further out, possibly there are only a finite number of twins with a spacing of exactly 2? Anyway, while I'm fairly sure the twin prime conjecture will one day be shown to be true, it can still be interesting to search for larger and larger twin prime pairs. The largest such known pair at the moment is
2996863034895*2^1290000 +/- 1
This is a pair with 388342 decimal digits. And while seriously large, it is still in range of large integers we can work with in MATLAB, though certainly not in double precision. In my own personal work on my own computer, I've done prime testing on integers (in MATLAB) with considerably more than 100,000 decimal digits.
But, again you may ask, just how does roughness help us here? In fact, this application of roughness is not new with me. You might want to read about tools like NewPGen {https://t5k.org/programs/NewPGen/} which sieves out numbers known to be composite, before any direct tests for primality are performed.
Before we even try to talk about numbers with thousands or hundreds of thousands of decimal digits, look at 6=2*3. You might observe
isprime([-1,1] + 6)
shows that both 5 and 7 are prime. This should not be a surprise, but think about what happens, about why it generated a twin prime pair. 6 is divisible by both 2 and 3, so neither 5 or 7 can possibly be divisible by either small prime as they are one more or one less than a multiple of both 2 and 3. We can try this again, pushing the limits just a bit.
isprime([-1,1] + 2*3*5)
That is again interesting. 30=2*3*5 is evenly divisible by 2, 3, and 5. The result is both 29 and 31 are prime, because adding 1 or subtracting 1 from a multiple of 2, 3, or 5 will always result in a number that is not divisible by any of those small primes. The next larger prime after 5 is 7, but it cannot be a factor of 29 or 31, since it is greater than both sqrt(29) and sqrt(31).
We have quite efficiently found another twin prime pair. Can we take this a step further? 210=2*3*5*7 is the smallest such highly composite number that is divisible by all primes up to 7. Can we use the same trick once more?
isprime([-1,1] + 2*3*5*7)
And here the trick fails, because 209=11*19 is not in fact prime. However, can we use the large twin prime trick we saw before? Consider numbers of the form [-1,1]+a*210, where a is itself some small integer?
a = 2;
isprime([-1,1] + a*2*3*5*7)
I did not need to look far, only out to a=2, because both 419 and 421 are prime. You might argue we have formed a twin prime "factory", of sorts. Next, I'll go out as far as the product of all primes not exceeding 60. This is a number with 22 decimal digits, already too large to represent as a double, or even as uint64.
prod(sym(primes(60)))
a = find(all(isprime([-1;1] + prod(sym(primes(60)))*(1:100)),1))
That easily identifies 3 such twin prime pairs, each of which has roughly 23 decimal digits, each of which have the form a*1922760350154212639070+/-1. The twin prime factory is still working well. Going further out to integers with 37 decimal digits, we can easily find two more such pairs that employ the product of all primes not exceeding 100.
prod(sym(primes(100)))
a = find(all(isprime([-1;1] + prod(sym(primes(100)))*(1:100)),1))
This is in fact an efficient way of identifying large twin prime pairs, because it chooses a massively composite number as the product of many distinct small primes. Adding or subtracting 1 from such a number will result always in a rough number, not divisible by any of the primes employed. With a little more CPU time expended, now working with numbers with over 1000 decimal digits, I will claim this next pair forms a twin prime pair, and is the smallest such pair we can generate in this way from the product of the primes not exceeding 2500.
isprime(7826*prod(sym(primes(2500))) + [-1 1])
ans =
logical
1
Unfortunately, 1000 decimal digits is at or near the limit of what the sym/isprime tool can do for us. It does beg the question, asking if there are alternatives to the sym/isprime tool, as an isProbablePrime test, usually based on Miller-Rabin is often employed. But this is gist for yet another set of posts.
Anyway, I've done a search for primes of the form
a*prod(sym(primes(10000))) +/- 1
having gone out as far as a = 600000, with no success as of yet. (My estimate is I will find a pair by the time I get near 5e6 for a.) Anyway, if others can find a better way to search for large twin primes in MATLAB, or if you know of a larger twin prime pair of this extended form, feel free to chime in.
What is a rough number? What can they be used for? Today I'll take you down a journey into the land of prime numbers (in MATLAB). But remember that a journey is not always about your destination, but about what you learn along the way. And so, while this will be all about primes, and specifically large primes, before we get there we need some background. That will start with rough numbers.
Rough numbers are what I would describe as wannabe primes. Almost primes, and even sometimes prime, but often not prime. They could've been prime, but may not quite make it to the top. (If you are thinking of Marlon Brando here, telling us he "could've been a contender", you are on the right track.)
Mathematically, we could call a number k-rough if it is evenly divisible by no prime smaller than k. (Some authors will use the term k-rough to denote a number where the smallest prime factor is GREATER than k. The difference here is a minor one, and inconsequential for my purposes.) And there are also smooth numbers, numerical antagonists to the rough ones, those numbers with only small prime factors. They are not relevant to the topic today, even though smooth numbers are terribly valuable tools in mathematics. Please forward my apologies to the smooth numbers.
Have you seen rough numbers in use before? Probably so, at least if you ever learned about the sieve of Eratosthenes for prime numbers, though probably the concept of roughness was never explicitly discussed at the time. The sieve is simple. Suppose you wanted a list of all primes less than 100? (Without using the primes function itself.)
% simple sieve of Eratosthenes
Nmax = 100;
N = true(1,Nmax); % A boolean vector which when done, will indicate primes
N(1) = false; % 1 is not a prime by definition
nextP = find(N,1,'first'); % the first prime is 2
while nextP <= sqrt(Nmax)
% flag multiples of nextP as not prime
N(nextP*nextP:nextP:end) = false;
% find the first element after nextP that remains true
nextP = nextP + find(N(nextP+1:end),1,'first');
end
primeList = find(N)
Indeed, that is the set of all 25 primes not exceeding 100. If you think about how the sieve worked, it first found 2 is prime. Then it discarded all integer multiples of 2. The first element after 2 that remains as true is 3. 3 is of course the second prime. At each pass through the loop, the true elements that remain correspond to numbers which are becoming more and more rough. By the time we have eliminated all multiples of 2, 3, 5, and finally 7, everything else that remains below 100 must be prime! The next prime on the list we would find is 11, but we have already removed all multiples of 11 that do not exceed 100, since 11^2=121. For example, 77 is 11*7, but we already removed it, because 77 is a multiple of 7.
Such a simple sieve to find primes is great for small primes. However is not remotely useful in terms of finding primes with many thousands or even millions of decimal digits. And that is where I want to go, eventually. So how might we use roughness in a useful way? You can think of roughness as a way to increase the relative density of primes. That is, all primes are rough numbers. In fact, they are maximally rough. But not all rough numbers are primes. We might think of roughness as a necessary, but not sufficient condition to be prime.
How many primes lie in the interval [1e6,2e6]?
numel(primes(2e6)) - numel(primes(1e6))
There are 70435 primes greater than 1e6, but less than 2e6. Given there are 1 million natural numbers in that set, roughly 7% of those numbers were prime. Next, how many 100-rough numbers lie in that same interval?
N = (1e6:2e6)';
roughInd = all(mod(N,primes(100)) > 0,2);
sum(roughInd)
That is, there are 120571 100-rough numbers in that interval, but all those 70435 primes form a subset of the 100-rough numbers. What does this tell us? Of the 1 million numbers in that interval, approximately 12% of them were 100-rough, but 58% of the rough set were prime.
The point being, if we can efficiently identify a number as being rough, then we can substantially increase the chance it is also prime. Roughness in this sense is a prime densifier. (Is that even a word? It is now.) If we can reduce the number of times we need to perform an explicit isprime test, that will gain greatly because a direct test for primality is often quite costly in CPU time, at least on really large numbers.
In my next post, I'll show some ways we can employ rough numbers to look for some large primes.
I've been trying this problem a lot of time and i don't understand why my solution doesnt't work.
In 4 tests i get the error Assertion failed but when i run the code myself i get the diag and antidiag correctly.
function [diag_elements, antidg_elements] = your_fcn_name(x)
[m, n] = size(x);
% Inicializar los vectores de la diagonal y la anti-diagonal
diag_elements = zeros(1, min(m, n));
antidg_elements = zeros(1, min(m, n));
% Extraer los elementos de la diagonal
for i = 1:min(m, n)
diag_elements(i) = x(i, i);
end
% Extraer los elementos de la anti-diagonal
for i = 1:min(m, n)
antidg_elements(i) = x(m-i+1, i);
end
end
I love it all
49%
Love the first snowfall only
11%
Hate it
19%
It doesn't snow where I live
22%
37 票
Los invito a conocer el libro "Sistemas dinámicos en contexto: Modelación matemática, simulación, estimación y control con MATLAB", el cual estará disponible pronto en formato digital.
El libro integra diversos temas de los sistemas dinámicos desde un punto de vista práctico utilizando programas de MATLAB y simulaciones en Simulink y utilizando métodos numéricos (ver enlace). Existe mucho material en el blog del libro con posibilidades para comentarios, propuestas y correcciones. Resalto los casos de estudio
Creo que el libro les puede dar un buen panorama del área con la posibilidad de experimentar de manera interactiva con todo el material de MATLAB disponible en formato Live Script. Lo mejor es que se pueden formular preguntas en el blog y hacer propuestas al autor de ejercicios resueltos.
Son bienvenidos los comentarios, sugerencias y correcciones al texto.
MATLAB FEX(MATLAB File Exchange) should support Markdown syntax for writing. In recent years, many open-source community documentation platforms, such as GitHub, have generally supported Markdown. MATLAB is also gradually improving its support for Markdown syntax. However, when directly uploading files to the MATLAB FEX community and preparing to write an overview, the outdated document format buttons are still present. Even when directly uploading a Markdown document, it cannot be rendered. We hope the community can support Markdown syntax!
BTW，I know that open-source Markdown writing on GitHub and linking to MATLAB FEX is feasible, but this is a workaround. It would be even better if direct native support were available.
I am very pleased to share my book, with coauthors Professor Richard Davis and Associate Professor Sam Toan, titled "Chemical Engineering Analysis and Optimization Using MATLAB" published by Wiley: https://www.wiley.com/en-us/Chemical+Engineering+Analysis+and+Optimization+Using+MATLAB-p-9781394205363
Also in The MathWorks Book Program:
Chemical Engineering Analysis and Optimization Using MATLAB® introduces cutting-edge, highly in-demand skills in computer-aided design and optimization. With a focus on chemical engineering analysis, the book uses the MATLAB platform to develop reader skills in programming, modeling, and more. It provides an overview of some of the most essential tools in modern engineering design.
Chemical Engineering Analysis and Optimization Using MATLAB® readers will also find:
- Case studies for developing specific skills in MATLAB and beyond
- Examples of code both within the text and on a companion website
- End-of-chapter problems with an accompanying solutions manual for instructors
This textbook is ideal for advanced undergraduate and graduate students in chemical engineering and related disciplines, as well as professionals with backgrounds in engineering design.
My following code works running Matlab 2024b for all test cases. However, 3 of 7 tests fail (#1, #4, & #5) the QWERTY Shift Encoder problem. Any ideas what I am missing?
Thanks in advance.
keyboardMap1 = {'qwertyuiop[;'; 'asdfghjkl;'; 'zxcvbnm,'};
keyboardMap2 = {'QWERTYUIOP{'; 'ASDFGHJKL:'; 'ZXCVBNM<'};
if length(s) == 0
se = s;
end
for i = 1:length(s)
if double(s(i)) >= 65 && s(i) <= 90
row = 1;
col = 1;
while ~strcmp(s(i), keyboardMap2{row}(col))
if col < length(keyboardMap2{row})
col = col + 1;
else
row = row + 1;
col = 1;
end
end
se(i) = keyboardMap2{row}(col + 1);
elseif double(s(i)) >= 97 && s(i) <= 122
row = 1;
col = 1;
while ~strcmp(s(i), keyboardMap1{row}(col))
if col < length(keyboardMap1{row})
col = col + 1;
else
row = row + 1;
col = 1;
end
end
se(i) = keyboardMap1{row}(col + 1);
else
se(i) = s(i);
end
% if ~(s(i) = 65 && s(i) <= 90) && ~(s(i) >= 97 && s(i) <= 122)
% se(i) = s(i);
% end
end
Too small
22%
Just right
38%
Too large
40%
2648 票
In one of my MATLAB projects, I want to add a button to an existing axes toolbar. The function for doing this is axtoolbarbtn:
axtoolbarbtn(tb,style,Name=Value)
However, I have found that the existing interfaces and behavior make it quite awkward to accomplish this task.
Here are my observations.
Adding a Button to the Default Axes Toolbar Is Unsupported
plot(1:10)
ax = gca;
tb = ax.Toolbar
Calling axtoolbarbtn on ax results in an error:
>> axtoolbarbtn(tb,"state")
Error using axtoolbarbtn (line 77)
Modifying the default axes toolbar is not supported.
Default Axes Toolbar Can't Be Distinguished from an Empty Toolbar
The Children property of the default axes toolbar is empty. Thus, it appears programmatically to have no buttons, just like an empty toolbar created by axtoolbar.
cla
plot(1:10)
ax = gca;
tb = ax.Toolbar;
tb.Children
ans = 0x0 empty GraphicsPlaceholder array.
tb2 = axtoolbar(ax);
tb2.Children
ans = 0x0 empty GraphicsPlaceholder array.
A Workaround
An empty axes toolbar seems to have no use except to initalize a toolbar before immediately adding buttons to it. Therefore, it seems reasonable to assume that an axes toolbar that appears to be empty is really the default toolbar. While we can't add buttons to the default axes toolbar, we can create a new toolbar that has all the same buttons as the default one, using axtoolbar("default"). And then we can add buttons to the new toolbar.
That observation leads to this workaround:
tb = ax.Toolbar;
if isempty(tb.Children)
% Assume tb is the default axes toolbar. Recreate
% it with the default buttons so that we can add a new
% button.
tb = axtoolbar(ax,"default");
end
btn = axtoolbarbtn(tb);
% Then set up the button as desired (icon, callback,
% etc.) by setting its properties.
As workarounds go, it's not horrible. It just seems a shame to have to delete and then recreate a toolbar just to be able to add a button to it.
The worst part about the workaround is that it is so not obvious. It took me a long time of experimentation to figure it out, including briefly giving it up as seemingly impossible.
The documentation for axtoolbarbtn avoids the issue. The most obvious example to write for axtoolbarbtn would be the first thing every user of it will try: add a toolbar button to the toolbar that gets created automatically in every call to plot. The doc page doesn't include that example, of course, because it wouldn't work.
My Request
I like the axes toolbar concept and the axes interactivity that it promotes, and I think the programming interface design is mostly effective. My request to MathWorks is to modify this interface to smooth out the behavior discontinuity of the default axes toolbar, with an eye towards satisfying (and documenting) the general use case that I've described here.
One possible function design solution is to make the default axes toolbar look and behave like the toolbar created by axtoolbar("default"), so that it has Children and so it is modifiable.
I am curious as to how my goal can be accomplished in Matlab.
The present APP called "Matching Network Designer" works quite well, but it is limited to a single section of a "PI", a "TEE", or an "L" topology circuit.
This limits the bandwidth capability of the APP when the intended use is to create an amplifier design intended for wider bandwidth projects.
I am requesting that a "Broadband Matching Network Designer" APP be developed by you, the MathWorks support team.
One suggestion from me is to be able to cascade a second section (or "pole") to the first.
Then the resulting topology would be capable of achieving that wider bandwidth of the microwave amplifier project where it would be later used with the transistor output and input matching networks.
Instead of limiting the APP to a single frequency, the entire s parameter file would be used as an input.
The APP would convert the polar s parameters to rectangular scaler complex impedances that you already use.
At that point, having started out with the first initial center frequency, the other frequencies both greater than and less than the center would come into use by an optimization of the circuit elements.
I'm hoping that you will be able to take on this project.
I can include an attachment of such a Matching Network Designer APP that you presently have if you like.
That network is centered at 10 GHz.
Kimberly Renee Alvarez.
310-367-5768
私の場合、前の会社が音楽認識アプリの会社で、アルゴリズム開発でFFTが使われていたことがきっかけでした。でも、MATLABのすごさが分かったのは、機械学習のオンライン講座で、Andrew Ngが、線型代数を使うと、数式と非常に近い構文のコードで問題が処理できることを学んだ時でした。
Overview
Authors:
- Narayanaswamy P.R. Iyer
- Provides Simulink models for various PWM techniques used for inverters
- Presents vector and direct torque control of inverter-fed AC drives and fuzzy logic control of converter-fed AC drives
- Includes examples, case studies, source codes of models, and model projects from all the chapters.
About this book
Successful development of power electronic converters and converter-fed electric drives involves system modeling, analyzing the output voltage, current, electromagnetic torque, and machine speed, and making necessary design changes before hardware implementation. Inverters and AC Drives: Control, Modeling, and Simulation Using Simulink offers readers Simulink models for single, multi-triangle carrier, selective harmonic elimination, and space vector PWM techniques for three-phase two-level, multi-level (including modular multi-level), Z-source, Quasi Z-source, switched inductor, switched capacitor and diode assisted extended boost inverters, six-step inverter-fed permanent magnet synchronous motor (PMSM), brushless DC motor (BLDCM) and induction motor (IM) drives, vector-controlled PMSM, IM drives, direct torque-controlled inverter-fed IM drives, and fuzzy logic controlled converter-fed AC drives with several examples and case studies. Appendices in the book include source codes for all relevant models, model projects, and answers to selected model projects from all chapters.
This textbook will be a valuable resource for upper-level undergraduate and graduate students in electrical and electronics engineering, power electronics, and AC drives. It is also a hands-on reference for practicing engineers and researchers in these areas.
I want to share a new book "Introduction to Digital Control - An Integrated Approach, Springer, 2024" available through https://link.springer.com/book/10.1007/978-3-031-66830-2.
This textbook presents an integrated approach to digital (discrete-time) control systems covering analysis, design, simulation, and real-time implementation through relevant hardware and software platforms. Topics related to discrete-time control systems include z-transform, inverse z-transform, sampling and reconstruction, open- and closed-loop system characteristics, steady-state accuracy for different system types and input functions, stability analysis in z-domain-Jury’s test, bilinear transformation from z- to w-domain, stability analysis in w-domain- Routh-Hurwitz criterion, root locus techniques in z-domain, frequency domain analysis in w-domain, control system specifications in time- and frequency- domains, design of controllers – PI, PD, PID, phase-lag, phase-lead, phase-lag-lead using time- and frequency-domain specifications, state-space methods- controllability and observability, pole placement controllers, design of observers (estimators) - full-order prediction, reduced-order, and current observers, system identification, optimal control- linear quadratic regulator (LQR), linear quadratic Gaussian (LQG) estimator (Kalman filter), implementation of controllers, and laboratory experiments for validation of analysis and design techniques on real laboratory scale hardware modules. Both single-input single-output (SISO) and multi-input multi-output (MIMO) systems are covered. Software platform of MATLAB/Simlink is used for analysis, design, and simulation and hardware/software platforms of National Instruments (NI)/LabVIEW are used for implementation and validation of analysis and design of digital control systems. Demonstrating the use of an integrated approach to cover interdisciplinary topics of digital control, emphasizing theoretical background, validation through analysis, simulation, and implementation in physical laboratory experiments, the book is ideal for students of engineering and applied science across in a range of concentrations.
I am excited to share my new book "Introduction to Mechatronics - An Integrated Approach, Springer, 2023" available through https://link.springer.com/book/10.1007/978-3-031-29320-7.
This textbook presents mechatronics through an integrated approach covering instrumentation, circuits and electronics, computer-based data acquisition and analysis, analog and digital signal processing, sensors, actuators, digital logic circuits, microcontroller programming and interfacing. The use of computer programming is emphasized throughout the text, and includes MATLAB for system modeling, simulation, and analysis; LabVIEW for data acquisition and signal processing; and C++ for Arduino-based microcontroller programming and interfacing. The book provides numerous examples along with appropriate program codes, for simulation and analysis, that are discussed in detail to illustrate the concepts covered in each section. The book also includes the illustration of theoretical concepts through the virtual simulation platform Tinkercad to provide students virtual lab experience.
I had originally planned on publishing my book via a traditional publisher, but am now reconsidering whether to use Amazon.com. I use Matlab and Latex in my book. It appears that it is not possible to publish is with Amazon due to this. Advice? Thanks. Kevin Passino
Hello,
Now that the "Copilot+PC" (Windows ARM) laptops are rapidly increasing in market share (Microsoft Surface Laptop, Dell XPS 13, HP OmniBook X 14, and more), are there any plans to provide builds for Matlab on Windows arm64?
Since there are already Windows builds of Matlab, it shouldn't be too hard to compile for Windows arm64, as far as I know. But I am not famaliar with Matlab's codebase.
Please try to publish Windows arm64 builds soon so that Matlab can be much more usable on Windows on ARM as it will run natively instead of in emulation.
Thank you very much.
Toolbox 全部入りの MATLAB ライセンス
67%
まだ持っていない Toolbox (下記にコメントください）
0%
MATLAB T シャツ
17%
MATLAB ルービックキューブ
0%
MATLAB 靴下
6%
MathWorks オフィス訪問チケット
11%
18 票
この場は MATLAB や Simulink を使っている皆さんが、気軽に質問や情報交換ができる場所として作られました。日本語でも気軽に投稿ができるように今回日本語チャネルを解説します。
ユーザーの皆様とのやり取りを通じて、みんなで知識や経験を共有し、一緒にスキルアップしていきましょう。 どうぞお気軽にご参加ください。
そして日本語チャネル開設にあたってコメントくださった皆様、ありがとうございます！
If you have a folder with an enormous number of files and want to use the uigetfile function to select specific files, you may have noticed a significant delay in displaying the file list.
Thanks to the assistance from MathWorks support, an interesting behavior was observed.
For example, if a folder such as Z:\Folder1\Folder2\data contains approximately 2 million files, and you attempt to use uigetfile to access files with a specific extension (e.g., *.ext), the following behavior occurs:
Method 1: This takes minutes to show me the list of all files
[FileName, PathName] = uigetfile('Z:\Folder1\Folder2\data\*.ext', 'File selection');
Method 2: This takes less than a second to display all files.
[FileName, PathName] = uigetfile('*.ext', 'File selection','Z:\Folder1\Folder2\data');
Method 3: This method also takes minutes to display the file list. What is intertesting is that this method is the same as Method 2, except that a file seperator "\" is added at the end of the folder string.
[FileName, PathName] = uigetfile('*.ext', 'File selection','Z:\Folder1\Folder2\data\');
I was informed that the Mathworks development team has been informed of this strange behaviour.
I am using 2023a, but think this should be the same for newer versions.
また、以下のリストから Web サイトを選択することもできます。
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