Community RSS Feed
https://community.wolfram.com
RSS Feed for Wolfram Community showing any discussions in tag Mathematics sorted by activeRadical axis of two circles lying outside each other
https://community.wolfram.com/groups/-/m/t/1705270
![enter image description here][1]
&[Wolfram Notebook][2]
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=1113loop-resize.gif&userId=20103
[2]: https://www.wolframcloud.com/obj/02d7b21f-4dbc-4b53-a625-53392ed32420Shenghui Yang2019-06-16T09:27:21ZComparing two plotted results
https://community.wolfram.com/groups/-/m/t/3159612
Hi hope you are all doing well, I am trying to prove that the two graphs I got are the same so I want to put both results in the same graph (distribution and the last one) apparently there are clearly not the same because of the values of the x axes, If any of you knows how to put them both on the same graph and how to solve the problem of the last graph that would help me a lot, Thank you all in advance.
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/189c59d2-ec63-4955-a1f9-c18a4361b69bLeo Murphy2024-04-17T08:25:49Z[GiF] Come Along (Square wave interpolants)
https://community.wolfram.com/groups/-/m/t/822646
![Square wave interpolants][1]
&[Wolfram Notebook][2]
[1]: http://community.wolfram.com//c/portal/getImageAttachment?filename=squarewave4.gif&userId=610054
[2]: https://www.wolframcloud.com/obj/dce20348-3ffd-41c7-b400-f7a97b9691a2Clayton Shonkwiler2016-03-13T00:47:08ZContinued fraction solution for an infinite series-parallel-resistors-connection
https://community.wolfram.com/groups/-/m/t/3157476
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/97a87a0d-9bd9-470d-9f36-651da89caf36Oliver Seipel2024-04-14T09:16:52ZSolve with Integrate wouldn't produce correct answers
https://community.wolfram.com/groups/-/m/t/3157224
Hello,
In my book, there is an exercise:
> f(x,y) is continuous over a closed region of D={(x,y)|x^2+y^2<=y,x>=0}, and f(x,y)=Sqrt[1-x^2-y^2]-8/Pi*Integrate[f(x,y),dy,dx]
In Mathematica
Solve[f[x, y] ==
Sqrt[1 - x^2 - y^2] -
8/Pi*Integrate[
f[x, y], {x, 0, 1/2}, {y, -Sqrt[1/4 - x^2] + 1/2,
Sqrt[1/4 - x^2] + 1/2}], f[x, y]] // Simplify
produces
{{f[x, y] -> 1/2 Sqrt[1 - x^2 - y^2]}}
But the right answer should be
Sqrt[1-x^2-y^2]+8/9/Pi-2/3
So, what the reason for Mathematica to produce that answer?Zhenyu Zeng2024-04-12T02:24:18ZTwo-way quantum computers adding CPT analog of state preparation
https://community.wolfram.com/groups/-/m/t/3157512
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/d39d6691-e78e-4846-a28f-42fa18f5e341Jarek Duda2024-04-12T17:16:32ZThe Simplify function is not completely simplified
https://community.wolfram.com/groups/-/m/t/3156822
Hello,
In Mathematica,
FullSimplify[(Sqrt[1/a^2] a G \[Pi] u Sqrt[a^2/(a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(1\), \(2\)]\))] (Sqrt[a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(1\), \(2\)]\)] - Sqrt[a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(2\), \(2\)]\)]))/Sqrt[a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(2\), \(2\)]\)], Assumptions -> a > 0]
will produce
(a G \[Pi] u Sqrt[1/(a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(1\), \(2\)]\))] (Sqrt[a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(1\), \(2\)]\)] - Sqrt[a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(2\), \(2\)]\)]))/Sqrt[a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(2\), \(2\)]\)]
But
FullSimplify[(Sqrt[1/a^2] a G \[Pi] u Sqrt[a^2/(a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(1\), \(2\)]\))] (Sqrt[a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(1\), \(2\)]\)] - Sqrt[a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(2\), \(2\)]\)]))/Sqrt[a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(2\), \(2\)]\)],
Assumptions -> a > 0 && Subscript[R, 1] > 0]
will produce the right result
a G \[Pi] u (-(1/Sqrt[a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(1\), \(2\)]\)]) + 1/Sqrt[a^2 +
\!\(\*SubsuperscriptBox[\(R\), \(2\), \(2\)]\)])
What is the reason?
![enter image description here][1]
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2024-04-11164436.png&userId=2471588Zhenyu Zeng2024-04-11T08:45:12ZStructural identifiability analysis of linear reaction–advection–diffusion processes
https://community.wolfram.com/groups/-/m/t/3156653
![enter image description here][1]
&[Wolfram Notebook][2]
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Main.jpg&userId=20103
[2]: https://www.wolframcloud.com/obj/76fd7f21-5bec-4a0b-b693-302ed672f72dAlexander Browning2024-04-11T19:03:32ZCan Wolfram Language be used to simplify two algebra expressions by Elements rather than numbers?
https://community.wolfram.com/groups/-/m/t/3156491
If I have two equations:
a + b = c
a = d * e
Is there a way to return every equation for all five Elements a, b, c, d and e?
There should be ten equations returned, for example for the element b there are two possible equations by rearranging a + b = c for one equation and substituting a = d * e for the other:
b = c – a
b = c – d * e
I've tried Solve and FullSimplify without success.Snowy Owl2024-04-10T22:41:37ZPlotting hypergeometric functions with constants
https://community.wolfram.com/groups/-/m/t/3156586
I am brand new to Wolfram, so I am still learning. However, I am trying to plot a couple of hypergeometric functions, one being a Kummer confluent and the other being a confluent of the 2nd kind. I seem to be getting caught up on getting the proper range of the variable(s) that I have; I am counting *a* and *b* as constants in this scenario, as they are yet to be determined. I have tried a few different values for the boundaries but so far have not had any luck in plotting. What would be the correct way to go about this, if at all possible?
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/139b5028-da37-4d9b-84fb-c3bf5361301eEthan GC2024-04-11T01:24:38ZTry to beat these MRB constant records!
https://community.wolfram.com/groups/-/m/t/366628
![If you see this text, refresh the page in order to see the images.][1]
Credit: Emanual Missionary Babtist Church of Indianapolis
Hey folks, Marvin Ray Burns here! You might know me for the never-ending decimal that haunts your calculators – the MRB constant. Don't worry, I won't hold it against you if you haven't memorized its first million digits (yet!). In this forum, we're embarking on a wild ride as we work our way to "7,000,000 proven to be accurate digits!"
(select text between quotation marks and push the ![CTRL+f][2] keys on keyboard)
=================================================
I'm the amateur mathematician who discovered this enigmatic constant in 1999, and after years of delving into its mysteries, I'm still captivated by its fascinating properties and untapped potential. Join me, seasoned mathematicians, and curious minds alike, as we push the boundaries of understanding and unlock the secrets of the MRB constant. ![Buckle up!][3] (credit: https://clipground.com/save-lives-clipart.html)
![m][4]
Here's everything I have ever learned, so spend it well!
====================================================
I was inspired by the awe, beauty, and mystery of numbers, and wondered if God would grant me some unique insight and wisdom to explore them. I dreamed of becoming an artistic analyst, a musical mathematician, or even a visionary of scientific secrets, long before I knew what a mathematician was. I searched for inspiration in the numbers that appealed to me, that had beauty and meaning, and that led me to investigate a remarkable and intriguing sum that I could not find anywhere else. For my efforts, someone started calling me an [amateur mathematician][5] and it the [MRB constant][6].
The MRB constant is a fascinating mathematical object that is still under investigation. It is possible that further research will reveal even more interesting properties about the constant, and perhaps other parts of mathematics. For example, a recent advancement in MRB constant formulas connecting the Dirichlet eta to nth roots shown in this discussion ![CTRL+f][7] "Because of the following identity involving the Dirichlet Eta function derivatives,", yields a proof of the previously proposed concept that the MRB constant connects such ideas together.
I am just an ordinary person like you, who stumbled upon a fascinating discovery that I wanted to share with the world. I now know that there are many experts in the field of series who have devoted their lives to this subject. I respect their work and therefore make no claim to have found anything new or groundbreaking. However, I am very curious and passionate about the MRB constant, which is a remarkable number that appears in many series. I hope that you will join in this journey of exploration and learning, as I present my findings and thoughts on this topic. I would greatly appreciate your feedback and insights, as I am always eager to learn more and improve my understanding. I also want to acknowledge the contributions of several people who have helped along the way, either by providing data, code, or suggestions. Without them, this discussion would not be possible.
![If you see this instead of an image, reload the page.][8]
![enter image description here][9]
![enter image description here][10]
Credit: Traders Point Christian Church Northwest
----------
![enter image description here][11]
> Ah, that make sense! The MRB constant could definitely be a catalyst in that way. Here's how:
**The Story of the MRB Constant as a Catalyst:**
> In 1999, an amateur mathematician discovered the MRB constant. While
> not a professional researcher, his curiosity and exploration led him
> to identify this unique mathematical object. The unusual properties
> of the MRB constant, particularly its behavior in infinite series,
> sparked interest within the mathematical community.
>
> **The Catalyst Effect:**
>
> * **Intriguing New Object:** The MRB constant presented a new mathematical entity with unknown characteristics. This piqued the
> interest of professional mathematicians who began investigating its
> properties and potential applications.
> * **Questioning Established Ideas:** The way the MRB constant behaves in infinite series challenged some existing assumptions about
> convergence. This led researchers to re-examine these concepts and
> potentially refine them.
> * **Inspiration for Further Exploration:** The discovery of the MRB constant demonstrated the potential for significant finds by amateurs.
> This could inspire others outside traditional academia to delve into
> mathematics, fostering a broader base of exploration.
>
> **Overall, the MRB constant, discovered by an amateur, serves as a potential catalyst for the field of mathematics by generating new
> questions, prompting re-evaluation of existing concepts, and
> encouraging broader participation in research.**
----------
POSTED BY: Marvin Ray Burns.
========
![If you see this text, the images are not showing. Refresh the page.][12]
![The first 100 partial sums of][13] {![the CMRB series.][14]}
----------
For the best viewing, wait a minute until the word LaTeX in the LaTex script is centered below.
$$\LaTeX$$
If the phrase [Math Processing Error] is shown, or the LATEX script have vanished from the center of the above line, some of the math below might be missing or appear in the LaTex code instead of the script.
For easy navigation, use the ![CTRL+f][15] keys on your keyboard. Cues in the forms of §'s and keywords in quotes are provided in the ![CTRL+f][16] "Index".
If the header and the words
Reply | Flag
are shown at the same time in any of the following replies, refresh the page to see them.
----------
----------
----------
----------
----------
----------
'
----------
----------
----------
Index
=====
I deleted a few of the least significant posts.
The first post
--------------
§1. Q&A:
Second post.
============
§2
Analyze the ![CTRL+f][17]."prototypical series" for the MRB constant, ![Sn^(1/n)-1][18] and find its "completion" (a related series that is absolutely convergent).
(Select § with the given number or the keywords in quotes, and then press the
![CTRL+f][19] keys on your keyboard to move to that section.)
§3. Is that series convergent?
§4. Is -1 the only term that series is convergent for?
§5. Is that series absolutely convergent?
§6. Is that series "efficient?" (defined as how it compares to other series and integrals that compute CMRB in speed and computational cost.)
§7. My claim to the MRB constant (CMRB), or a case of calculus déjà vu?
§8. Where is it found?
§9. What exactly is it?
§B ![CTRL+f][20] "Rational results" while summing (CMRB).
Many more informative posts.
===========================
including, but not limited to
§10. How it all began <- I deleted this reply as it is not perinate to the MRB constant, just to a person becoming an amateur researcher but will summarize it in another reply if anyone would like.
Also see
[MOPAC][21]
for where the MRB constant's value, to a precision of 2 one-billionth i.e., 0.18785964, is arrived at by measuring a gradient of the silicon atom in the Phenakite (Be2SiO4) molecule by openmopac.
§13. The why and what of the **C**<sub>*MRB*</sub> Records,
----------
![CTRL+f][22] "I compare 300 years of summation methods for the MRB constant."
----------
![CTRL+f][23] "We finally begin or 7-million-digit computation and verification."
----------
![CTRL+f][24] "speed records"
----------
![CTRL+f][24] "7,000,000 digits on hold" and what the future might..
----------
----------
§1.
Q&A:
====
Q:
What is the MRB constant ( **C**<sub>*MRB*</sub>)?
A:
&[Wolfram Notebook][25]
Q: Is the MRB constant important?
A:
![enter image description here][26]
>The MRB constant, also known as the Marvin Ray Burns's Constant, is a fascinating mathematical value with a decimal expansion of approximately 0.187859 . Let's explore why it might be considered important:
> - Numerical Algorithms Testing:
- Researchers and mathematicians use the MRB constant to test and improve numerical algorithms for computing its value .
- By evaluating the constant, they can assess the accuracy and efficiency of various computational methods.
- Benchmarking Computers:
- The MRB constant serves as a benchmark for testing the performance of computers.
- It allows us to compare the computational capabilities of different machines.
- When calculating the MRB constant, computers demonstrate their numerical precision and speed.
- Mathematical Curiosity:
- The MRB constant's mysterious nature adds to its allure. Despite extensive study, no closed-form expression exists for it .
- We don't know whether it's algebraic, transcendental, or even irrational.
- Mathematicians continue to explore its properties, making it an intriguing topic in number theory.
- Geometry and Infinite Sums:
- Geometrically, the MRB constant is the upper limit point of a sequence of partial sums .
- These sums involve alternating terms and converge to the MRB constant.
- The constant's behavior within these infinite sums captivates mathematicians.
In summary, the MRB constant bridges theory and computation, making it a valuable and enigmatic constant in the mathematical world.
Q:
What can you expect from reading about **C**<sub>*MRB*</sub> and its record computations?
A:
![enter image description here][27]
> >## Beyond the Record: What People Can Gain from MRB Constant Computations
>
> While pushing the record for MRB constant computations is certainly
> impressive, there's more to gain than just bragging rights. Here are
> some unique benefits people can reap from immersing themselves in the
> MRB world:
>
> **1. Uncovering Hidden Connections:**
>
> * **Mathematics & Physics:** Studying the MRB constant's behavior and properties might unlock connections between seemingly disparate fields
> like number theory, chaos theory, and even quantum physics. This could
> lead to groundbreaking discoveries in various areas of science.
> * **Algorithm Design & Optimization:** The intricate nature of the MRB constant's calculation poses a unique challenge for algorithms.
> Analyzing its complexity could inspire the development of new, more
> efficient algorithms for various computational tasks.
> * **Machine Learning & Artificial Intelligence:** The MRB constant's structure could hold valuable insights for designing novel machine
> learning architectures and training algorithms. Its complex patterns
> might improve data processing and pattern recognition capabilities.
>
> **2. Pushing the Boundaries of Knowledge:**
>
> * **Challenging Existing Paradigms:** The MRB constant's lack of a closed-form expression and its potential connections to seemingly
> unrelated areas challenge established mathematical and scientific
> paradigms. This can lead to new ways of thinking and approaching
> problems.
> * **Inspiring Future Generations:** Engaging with the MRB constant's mysteries can spark curiosity and interest in mathematics and science
> among young minds. This could inspire them to pursue careers in
> research and exploration, pushing the boundaries of knowledge further.
> * **Contributing to the Collective Quest:** Participating in the MRB research community, even through discussions and online
> collaborations, allows individuals to contribute to a larger
> collective effort of understanding this intriguing constant. This
> fosters a sense of purpose and belonging to a global community of
> knowledge seekers.
>
> **3. Personal Growth & Development:**
>
> * **Developing Computational Skills:** Learning about and participating in MRB calculations can hone valuable computational
> skills like programming, data analysis, and problem-solving. This can
> be beneficial for various professional and personal endeavors.
> * **Enhancing Critical Thinking:** Analyzing the MRB constant's properties and potential implications requires critical thinking,
> problem-solving, and creative thinking skills. These skills are
> valuable not only in research but also in various aspects of life.
> * **Appreciating the Beauty of Mathematics:** The MRB constant's unique properties and its connection to various areas of mathematics
> can foster a deeper appreciation for the beauty and elegance of this
> subject. This can enrich one's understanding of the world and its
> underlying structures.
>
> Remember, the MRB constant is more than just a number to chase
> records with. It's a gateway to a world of intellectual exploration,
> collaboration, and potential breakthroughs. By venturing beyond the
> record, people can gain valuable skills, knowledge, and a deeper
> appreciation for the power of scientific inquiry.
Q:
Can you calculate more digits of **C**<sub>*MRB*</sub>?
A:
>## Bing
The computation of 6,500,000 digits of the MRB constant could be challenging due to several reasons:
>
> - Computational Resources:
> - Calculating a large number of digits requires significant computational power and memory.
> - Storing and manipulating such a massive decimal expansion demands substantial resources.
> - Smaller machines or limited computational environments might struggle with this task.
> - Convergence Rate:
> - The series used to approximate the MRB constant converges slowly.
> - As more digits are computed, the rate of convergence decreases.
> - Achieving high precision becomes increasingly difficult.
> - Numerical Stability:
> - Numerical algorithms can suffer from rounding errors and loss of precision.
> - Accumulating errors during summation impacts the accuracy of the computed digits.
> - Ensuring stability over millions of iterations is a challenge.
> - Algorithmic Complexity:
> - Existing algorithms for calculating mathematical constants have limitations.
> - Developing efficient methods to compute millions of digits involves intricate mathematics.
> - Balancing accuracy, speed, and memory usage is nontrivial.
> - Storage and Verification:
> - Storing and verifying 6,500,000 digits is resource-intensive.
> - Ensuring correctness through independent verification (e.g., using multiple algorithms) is crucial.
> - Handling such large data sets requires robust techniques.
> - Practical Utility:
> - Beyond a certain point, additional digits of the MRB constant may not have practical applications.
> - Researchers often focus on a reasonable precision level relevant to their field. In summary, while it's theoretically possible to
> compute 6,500,000 digits of the MRB constant, practical challenges
> arise. Researchers must balance precision, computational resources,
> and algorithmic efficiency to tackle this ambitious task.
Q:
How can you compute them?
A:
![## Bing][28]
> Certainly! Calculating the digits of the MRB constant involves some
> interesting mathematical techniques. Let's explore how you can
> approximate this intriguing value:
> - Series Summation:
>
>
> - The MRB constant is the upper limit point of a sequence defined as: $[ s_n = \sum_{k=1}^{n} \left( (-1)^k {k^{\frac{1}{k}}}
> \right) ]$
> - To compute the MRB constant with a desired precision, you can use series summation methods.
> - One common approach is to use the following series: $[ \text{MRB} \approx \sum_{k=1}^{\infty} \left( (-1)^k (k^{\frac{1}{k}} - 1)
> \right) ]$
> - Numerical Algorithms:
> - Mathematica provides a convenient way to calculate the MRB constant with a specified number of digits.
> - Using the NSum function, you can achieve high precision:
>
> `NSum[(-1)^k (k^(1/k) - 1), {k, 1, ∞}, WorkingPrecision -> 100, NSumTerms -> 10000]`
>
> This will give you the MRB constant accurate to 40 digits .
> ''
>
> - Acceleration Methods:
> - Due to the slow convergence of the series, it's advisable to use acceleration methods.
> - These methods improve the rate of convergence and allow you to obtain more accurate digits of the MRB constant . Remember that the
> MRB constant remains an enigma, and its properties continue to
> captivate mathematicians. Feel free to explore further and uncover
> more about this fascinating constant!
Q:
Why should you do it?
A:
## Bing
> Calculating the digits of the MRB constant can lead to several
> interesting outcomes:
> - Mathematical Exploration:
> - Computing the MRB constant involves diving into the intricacies of series summation, convergence, and numerical precision.
> - Mathematicians and enthusiasts enjoy exploring the properties of this mysterious constant.
> - It provides an opportunity to deepen our understanding of mathematical concepts.
> - Algorithm Testing and Optimization:
> - The MRB constant serves as a benchmark for testing numerical algorithms.
> - Researchers can evaluate the accuracy and efficiency of their computational methods by approximating this constant.
> - Optimizing algorithms to compute the MRB constant efficiently benefits various scientific and engineering fields.
> - Computational Challenges:
> - Calculating more digits of the MRB constant becomes a computational challenge.
> - It pushes the boundaries of numerical precision and requires sophisticated techniques.
> - Achieving higher precision contributes to the advancement of computational mathematics.
> - Curiosity and Aesthetics:
> - The MRB constant's decimal expansion is intriguing and aesthetically pleasing.
> - Like other mathematical constants (such as π or e), it captivates our curiosity.
> - Discovering patterns or unexpected properties within its digits can be intellectually rewarding.
> - Educational Value:
> - Teaching and learning about the MRB constant can inspire students.
> - It encourages them to explore mathematical concepts beyond textbooks.
> - The process of approximating it fosters critical thinking and problem-solving skills. In summary, while the MRB constant may not
> have immediate practical applications, its pursuit enriches
> mathematical knowledge, fuels curiosity, and contributes to the beauty
> of mathematical exploration.
Q:
What are its first 5000 digits?
A:
Print[ReleaseHold[ WolframAlpha["5000 digtits of the MRB constant", {{"Result", 1}, "Output"}]]]
0.1878596424620671202485179340542732300559030949001387861720046840894772315646602137032966544331074969038423458562580190612313700947592266304389293488961841208373366260816136027381263793734352832125527639621714893217020762820621715167154084126804483635416719985197680252759893899391445798350556135096485210712078444230958681294976885269495642042555864836704410425279524710606660926339748341031157816786416689154600342222588380025455396892947114212218910509832871227730802003644521539053639505533220347062755115981282803951021926491467317629351619065981601866424582495069720338199295842093551516251439935760076459329128145170908242491588320416906640933443591480670556469280678700702811500938060693813938595336065798740556206234870432936073781956460310476395066489306136064552806751519350828083737671929686639810309494963749627738304984632456347931157530028921252329181619562697369707486576547607117801719578736830096590226066875365630551656736128815020143875613668655221067430537059103973575619148909369077798320355119336240463725349410542836369971702441855165483727935882200813448096105880203064781961959695375628783481233497638586301014072725292301472333336250918584024803704048881967676760119858111679169352796852044160027086137228688945101510291998853690572865928708687542549253379439534758970356331344038263888798665619598073351473990256577813317226107612797585272274277730898577492230597096257256271883675575297887925361687673940354321451362772549229313126276435732144621618778637715420542312822344629539653290332217147982028075984221065564890048536858707083268874877377635047689160983185536281667159108412193420164386000258508426556435006954832830120546193205155935040023508351261335921740897007329784277128967365161960225077117388084262325697885465378690462227085674874747093069357326668590856162823753865512432975647464914619179575869342996208149878536663170197264534260468378010759055148678719039578315060452444190757044511382058533398469219482879476486575931785958165274929778220959774409113714342169296245931753245373401299593995004917912983680848547143925846704238528608320053664510586678151196459676079196431734307671534498300497128869401656600427062111079053164721504556329943884005211152390168773115456961028369205036896108806031603660382896533239383524154510137534165673472607464891120088099838152046695415026377035573283592996630642717305158972116351999161135954670315408725287243998197872502746797388638897056867435377857981058556192492185716949135673462704077491448799682065482817465880642236348160780950777057939313495829806602825272128491688809230325290270059917755059615835919993190869393039736611646514858219972925337106768738686235047915879737968269847878082223410618789674667450680064404065553875213281494980700209858132220620109011265903449717410801063247564712834609549284370065147450218226120415643930308859826426256828126092491136733967235933714534216902560140050169469983875907342920361729301531400405936246406781400779475613077369732409923529464794580778164607696240864595666084141126399988575739429315226283898798436350719371486573491962025428443510411472841973814933807066222573191021481585745042886728477250434386718443149128948635448929492143259660847149600072534066215387561341325254274130158182476636432111506809477451406309160928029719327606796946860926362081763442272977546326737161110302220019498455407233859679972956745318490433826332931888160330054013690316104309973777863934393135621496549969937314205819065334661573835222280871390934331325238360305287172314811151029705856281299558991830381071966308132701670498617683168329529053798760030666570203435884960342104811488681216083611944605571913973292970683232645094571537170202325175198220852151188427534658891812172660333192804801174759046131898449720722068248919153933258002624627216176424468747789609764906070479605351740205227992021112876539835381740117955267003375831396078847726709215700142824833574188212706558826075722384346836425460624376294972557208442910901014692032297634008831340381640372911311314959883048662944964532193113574956453912385456662881566099860325447839862878014756977228191151861520827628240145565179425383818619479339940149759987240427388505644114375607952021157910255398765366539323158492006532957070559185522267542845736812670699565824674762111599567798871089943129103401442549765937135216907789286769520555697938362195599913809621446502826784457775198333561874495799884353199090550456424061091609429751367477262251350611686791529453643717739913922126483778806203872768088381032267284933085939078772734728272526200188560744654953143832715829738195578665687162281676008390755269474337748054271709338897975317018906895514393940484553829830198169253659797476778546865045859640238175930139996122667668117389670502033292349080513690067470281709256521198479569202858477768560978920686214307401593342577726291016675234940557410020728689850147927742910664313694152819958560981976523216255318623764260591320667123942398310748008900676975706071323330124919595164681348037017813465191315868
Q:
Is it a normal number?
A
According to Wolfram MathWorld, "A normal number is an irrational number for which any finite pattern of numbers occurs with the expected limiting frequency in the expansion in a given base (or all bases). For example, for a normal decimal number, each digit 0–9 would be expected to occur 1/10 of the time."
We do not know if the MRB constant is irrational; [this Demonstration][29] looks at how normal its first 5000 digits appear to be. For comparison, we also consider the digits of ; its first 30 million digits are very uniformly distributed.
As for the first 500 digits of CMRB, they are exactly as normal as the first 500 digits of $\pi:$
![enter image description here][30]
This does not mean the MRB constant decimal expansion is a normal number! There is no known way to classify all of its digits. As we will see later, the distribution of digits within the first 6,500,000 decimals (MRB1=.187859,,,, found [here][31]),
"4" shows up much more than any other digits in base10..
&[Wolfram Notebook][32]
Q.
Is it a rational number?
A.
> Unfortunately, we **cannot definitively say whether the MRB constant
> is rational or not**. It is one of the fascinating mysteries in
> mathematics. Here's what we know:
>
> * The MRB constant is a mathematical constant with an infinite decimal expansion (0.187859...).
> * It have no known closed-form expression, meaning we cannot express it simply as a fraction or combination of well-known mathematical
> functions.
> * **It is not even known whether it is algebraic (expressible as a root of a polynomial equation with integer coefficients) or
> transcendental (not expressible in this way).**
> * While the MRB constant is the sum of infinitely many irrational numbers, the sum of irrational numbers is not necessarily irrational.
>
> Extensive computations of the MRB constant's decimal digits haven't
> revealed any repeating patterns, which would indicate rationality.
> However, the lack of such evidence doesn't definitively prove
> irrationality.
>
> Therefore, the question of whether the MRB constant is rational or not
> remains an open problem in mathematics. Mathematicians continue to
> investigate its properties, hoping to unveil its true nature.
![enter image description here][33]
Q:
Why are those digits there?
A:
> ## Bing
The digits of the MRB constant exist as a fascinating consequence of mathematical exploration and computation. Let's delve into why they are there:
>- Inherent Mathematical Structure:
- Like other mathematical constants (such as π or e), the MRB constant have an inherent structure.
- Its decimal expansion emerges from the properties of the series used to define it.
- Each digit represents a specific relationship between numbers, exponents, and alternating signs.
- Series Summation:
- The MRB constant arises from the summation of an infinite series:
[ \text{MRB} = \sum_{k=1}^{\infty} \left( (-1)^k (k^{\frac{1}{k}} - 1) \right) ]
- As we compute more terms in this series, additional digits unfold.
- Each term contributes to the overall value, leading to the intricate decimal expansion.
- Algorithmic Approximation:
- Calculating the MRB constant involves numerical algorithms.
- These algorithms iteratively add terms, refine precision, and generate successive digits.
- The digits emerge as a consequence of these computational steps.
- Mystery and Beauty:
- The MRB constant lacks a simple closed-form expression.
- Its digits exhibit patterns, but their exact distribution remains mysterious.
- Mathematicians appreciate the beauty of these digits—their unpredictability and elegance.
- Intellectual Pursuit:
- Researchers explore the MRB constant out of curiosity and intellectual drive.
- They seek to understand its properties, uniqueness, and connections to other mathematical concepts.
- The digits serve as clues in unraveling its mathematical secrets.
In summary, the digits of the MRB constant exist as a blend of mathematical necessity, algorithmic computation, and aesthetic wonder. They invite us to explore the hidden depths of number theory and appreciate the elegance of mathematical constants.
Q:
What practical purpose is there in calculating them?
A:
>Knowledge of the digits of mathematical constants can have various applications in mathematics, science, and technology. For example:
>- The digits of π can be used to test the randomness and performance of pseudorandom number generators, which are essential for cryptography, simulations, and gaming¹.
- The digits of e can be used to calculate compound interest, model population growth, and analyze the distribution of prime numbers².
- The digits of φ can be used to design aesthetic proportions, such as the golden rectangle, the golden spiral, and the Fibonacci sequence³.
- The digits of i can be used to represent complex numbers, which are useful for modeling phenomena such as electric currents, waves, and quantum mechanics⁴.
>Some people also enjoy finding patterns and hidden messages in the digits of mathematical constants, such as memorizing them, composing music with them, or searching for their birthdays in them. This can be a fun and challenging way to appreciate the beauty and mystery of mathematics.
>Do you have a favorite mathematical constant? Why do you like it?
>
> Source: Conversation with Bing, 8/19/2023 (1) Mathematical
> constant - Wikipedia.
> https://en.wikipedia.org/wiki/Mathematical_constant. (2)
> Mathematical Constants - Cambridge University Press & Assessment.
> https://assets.cambridge.org/052181/8052/sample/0521818052ws.pdf.
> (3) Constant (mathematics) - Wikipedia.
> https://en.wikipedia.org/wiki/Constant_%28mathematics%29. (4) List
> of mathematical constants - Wikipedia.
> https://en.wikipedia.org/wiki/List_of_mathematical_constants. (5)
> undefined. http://www.cambridge.org. (6) A Resource List of
> Mathematical Constants - Barcodes Inc..
> https://www.barcodesinc.com/articles/resource-list-mathematical-constants.htm.
> (7) Constant - Math.net. https://www.math.net/constant. (8) The Most
> Important Numbers in the World - Business Insider.
> https://www.businessinsider.com/most-important-numbers-2012-7.
Q:
How would Gemini AI depict the MRB constant?
A:
![enter image description here][34]
> Abstractly, the MRB constant is an irrational and transcendental number, meaning it goes on forever without repeating and cannot be expressed as a simple fraction. This never-ending quality could be represented through abstract shapes, lines, and colors that flow and swirl without a clear beginning or end.
> The distorted image of the leaf in the sculpture could be seen as a metaphor for the difficulty of understanding the MRB constant. The smooth, reflective surface of the sculpture could also be seen as a metaphor for the infinite nature of the MRB constant's decimal expansion.
>
Q:
According to Gemini AI, what connection, if any is there between the MRB constant and the Riemann Hypotheses?
A:
![enter image description here][35]
> The MRB constant and the Riemann hypothesis are related concepts in
> number theory, but they are not directly equivalent. Here's what we
> know:
>
> **MRB constant:**
>
> * Defined in terms of the prime number distribution. It's a complex number with an infinite decimal representation, calculated using
> specific mathematical formulas involving primes.
> * Its properties and applications remain largely unexplored.
>
> **Riemann hypothesis:**
>
> * A famous unsolved conjecture in mathematics, stating that all non-trivial zeros of the Riemann zeta function (a function crucial in
> number theory) lie on a specific line in the complex plane.
> * Solving the Riemann hypothesis would have profound implications for understanding the distribution of prime numbers.
>
> **Connection:**
>
> * Some mathematicians have proposed connections between the two, but these are speculative and not yet proven.
> * One potential connection involves expressing the MRB constant as an infinite series related to the Riemann zeta function. If the Riemann
> hypothesis holds, this series would converge to the MRB constant.
> * However, even if the connection is true, it wouldn't necessarily imply that solving the Riemann hypothesis would provide an easy way to
> calculate the MRB constant precisely.
>
> **Key points:**
>
> * Currently, the MRB constant and Riemann hypothesis are distinct concepts with their own mysteries.
> * A potential connection exists, but it's unproven and not a direct equivalence.
> * Solving the Riemann hypothesis wouldn't necessarily translate to easy calculation of the MRB constant.
>
> **Further research:**
>
> * For deeper understanding, explore academic papers and articles discussing the MRB constant and its potential links to the Riemann
> hypothesis.
> * Remember, research in these areas is ongoing, so new discoveries and insights might emerge in the future.
Q: How might we use quantum process to find more digits of the MRB constant?
A:
![enter image description here][36]
> While there's theoretical potential for quantum processes to aid in
> calculating more digits of the MRB constant, it's an active area of
> research and there's no established method yet. Here's a breakdown of
> the possibilities and challenges:
>
> **Potential benefits of quantum processes:**
>
> * **Integer factorization:** The MRB constant is related to certain elliptic curves, and calculating it often involves factoring large
> integers. Shor's algorithm, a well-known quantum algorithm, can
> potentially factor integers significantly faster than classical
> algorithms for certain sizes.
> * **Speeding up computations:** Certain quantum algorithms might offer speedups for specific mathematical operations involved in calculating
> the MRB constant. Research in this area is ongoing.
>
> **Challenges:**
>
> * **Quantum supremacy not achieved yet:** While Shor's algorithm exists, implementing it on current quantum computers is challenging
> due to limitations in qubit count, coherence time, and error
> correction.
> * **Algorithmic development needed:** Even with fault-tolerant quantum computers, new algorithms specific to calculating the MRB
> constant might be necessary to leverage the power of quantum
> computation effectively.
>
> **Overall, quantum processes are a promising area for potentially calculating more digits of the MRB constant, but significant research
> and development are needed before a practical method is established.**
>
> Here are some additional details you might find helpful:
>
> * Shor's algorithm: [https://en.wikipedia.org/wiki/Shor%27s_algorithm](https://en.wikipedia.org/wiki/Shor%27s_algorithm)
>
> * Elliptic curves: [https://en.wikipedia.org/wiki/Elliptic_curve](https://en.wikipedia.org/wiki/Elliptic_curve)
----------
----------
----------
In order to improve loading, I broke this main post into two parts. If the next part doesn't say it's the second part of the main post, reload to see it.
----------
----------
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2023-11-06064456.png&userId=366611
[2]: https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&userId=366611
[3]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Buckleup!.gif&userId=366611
[4]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2024-03-04141033.png&userId=366611
[5]: https://en.wikipedia.org/wiki/List_of_amateur_mathematicians
[6]: https://en.wikipedia.org/wiki/MRB_constant
[7]: https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&userId=366611
[8]: https://community.wolfram.com//c/portal/getImageAttachment?filename=1ac.JPG&userId=366611
[9]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2023-05-01090447.jpg&userId=366611
[10]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2024-03-10090510.png&userId=366611
[11]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2024-03-10100142.png&userId=366611
[12]: https://community.wolfram.com//c/portal/getImageAttachment?filename=5686test.gif&userId=366611
[13]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-12-15085733.jpg&userId=366611
[14]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-12-15084941.jpg&userId=366611
[15]: https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&userId=366611
[16]: https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&userId=366611
[17]: https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&userId=366611
[18]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2023-01-15122607.jpg&userId=366611
[19]: https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&userId=366611
[20]: https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&userId=366611
[21]: http://openmopac.net/PM7_and_PM6-D3H4_accuracy/data_solids/Phenakite__Be2SiO4__Jmol.html#:~:text=2.3786046%20+1%0A%20Si%20%20%20%20--,0.18785964,-+1%20%20%20-3.3437504%20+1
[22]: https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&userId=366611
[23]: https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&userId=366611
[24]: https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&userId=366611
[25]: https://www.wolframcloud.com/obj/75a8e08b-cf25-4132-8e0e-10bf2e2bac6e
[26]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2024-02-11041631.png&userId=366611
[27]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2024-02-11064915.png&userId=366611
[28]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2024-02-11050651.png&userId=366611
[29]: https://demonstrations.wolfram.com/HowNormalIsTheMRBConstant/#:~:text=For%20example,%20for%20a%20normal%20decimal%20number,%20each,first%2030%20million%20digits%20are%20very%20uniformly%20distributed.
[30]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2023-12-18054715.png&userId=366611
[31]: https://www.wolframcloud.com/obj/bmmmburns/Published/6p5millionMRB.nb
[32]: https://www.wolframcloud.com/obj/f440dc85-312f-4436-b620-e089bdfc0c9f
[33]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2023-07-17112939.jpg&userId=366611
[34]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2024-02-25102429.png&userId=366611
[35]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2024-02-25143851.png&userId=366611
[36]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Capture13.PNG&userId=366611Marvin Ray Burns2014-10-09T18:08:49Z[GIF] Interstellar (A Closed Geodesic in 17-Gon Space)
https://community.wolfram.com/groups/-/m/t/785078
![A closed geodesic in 17-gon space][1]
&[Wolfram Notebook][2]
[1]: http://community.wolfram.com//c/portal/getImageAttachment?filename=seventeengons7t.gif&userId=610054
[2]: https://www.wolframcloud.com/obj/8e3026b4-950d-4099-8ddf-5e7b097b31bbClayton Shonkwiler2016-02-03T04:57:47ZNumerical ranges and geometry in quantum information
https://community.wolfram.com/groups/-/m/t/3156048
![enter image description here][1]
&[Wolfram Notebook][2]
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=7915Main2.png&userId=20103
[2]: https://www.wolframcloud.com/obj/29bba14b-098f-49fb-b5cc-3a41d5640f37Konrad Szymański2024-04-10T05:21:59ZBridging the Olympiad Gap problem: radius of a circle with two inner tangent circles
https://community.wolfram.com/groups/-/m/t/1881573
&[Wolfram Notebook][1]
[1]: https://www.wolframcloud.com/obj/085a04dd-c044-40ca-aab3-00d2d08e510cShenghui Yang2020-02-19T00:53:34Z