On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication

Pushpendra Singh; Anubha Gupta; Shiv Dutt Joshi

doi:10.1371/journal.pone.0312502

Peer Review History

Original SubmissionJune 19, 2024
30 Aug 2024 Decision Letter - Sania Asif, Editor PONE-D-24-24879On the hypercomplex numbers and normed division algebra of all dimensions: A unified multiplicationPLOS ONE Dear Dr. Singh, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. Please submit your revised manuscript by Oct 14 2024 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. 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Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice. Additional Editor Comments: Manuscript ID: PONE-D-24-24879 Title: On the hypercomplex numbers and normed division algebra of all dimensions: A unified multiplication. Journal: PLOS ONE Dear Pushpendra Singh, Thanks for submitting your manuscreipt to the Journal, PLOS ONE. 1. Please revise your paper according to the referee's advice in the attachment. 2. Moreover, please check the whole paper carefully again to avoid typos and mistakes. Please do not hesitate to contact us if you have any questions regarding the revision of your manuscript or if you need more time. We look forward to hearing from you soon. Many thanks! Best regards Sania Asif [Note: HTML markup is below. Please do not edit.] Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #1: Yes Reviewer #2: Yes ******** 2. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: N/A Reviewer #2: Yes ****** 3. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #1: Yes Reviewer #2: Yes ****** 4. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: Yes Reviewer #2: Yes ****** 5. Review Comments to the Author Reviewer #1: pdf file attached with detailed comments pdf file attached with detailed comments Reviewer #2: In this paper the authors present a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Moreover, they elucidated the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Importantly, they explored its utility in point cloud image processing. The paper is well supported by the examples and figures which puts life to the paper. The paper is well written and fits to the scope of the journal. I recommend acceptance. ****** 6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #1: No Reviewer #2: No ******** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. 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Revision 1
18 Sep 2024 Author Response Dear Editor: First of all, please accept our thanks for devoting time and coordinating the reviews of the submitted manuscript Ref. No. PONE-D-24-24879 titled ``On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication'' for possible publication in PLOS ONE. We submitted the manuscript on June 19, 2024 and received the review comments from the reviewers on August 31, 2024. We appreciate the time and effort you and the anonymous reviewers devoted to reviewing and providing useful comments to help us improve the quality of the manuscript. We have done our best to address the comments of the reviewers. We, hereby, submit: (1) a response letter with detailed replies to each of the reviewers’ comments and (2) a revised manuscript that we believe to have significantly improved after incorporating suggestions of the reviewers. Please do thank the reviewers on our behalf for their time and effort. We also thank you for your work as the editor. We have also acknowledged editors and anonymous reviewers in the acknowledgments section of the revised manuscript. Thanks and regards, Authors Response to Reviewers: PONE-D-24-24879 Title: On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication Reviewer #1: Comments to authors The manuscript describes an extension of complex algebra to higher dimensions. Three dimensions are treated in detail. The proposal is very interesting and worth pursuing. Response: Authors would like to thank the reviewer for providing detailed comments in the previous review round that helped us improve the quality of the manuscript. Minor corrections: p.5 The spherical coordinate system is formally defined in terms of the trigonometric functions, in particular, $\\phi=\\arctan(\\frac{b}{a})$. It is only in computer science where `$\\mathrm{tan2}$' is used. It would be better to use the formal definition and, if required, introduce the computational definition thereafter. Response: As per suggestion of the reviewer, we have used $\\phi=\\arctan(\\frac{b}{a})$, and introduced the computational definition thereafter. p.5 Eq. (7) does not follow from Eq. (1)-(2). The wording before Eq. (7) has to be careful to avoid misunderstandings. Response: As per suggestion of the reviewer, we have changed text before (7) to avoid any misunderstandings. p.7 Addition and multiplication are the two fundamental operations. Subtraction and division come from inverse elements. Thus, it is customary to define only two operations. Response: As per suggestion of the reviewer, we have defined addition and multiplication as two fundamental operations, and mentioned that subtraction and division come from inverse elements. p.8 Figure 2 can be greatly improved. The trajectory is not clear in (b) and (c), where it begins/ends and which path it follows. In fact, maybe (c) could be deferred mentioning that the ZY plane and ZX planes exhibit similar behaviour for the appropriate values of $\\phi$. Response: Based on the suggestion of the reviewer, we have revised Figure 2, specifically parts (b) and (c). In these updated figures, the left semicircle is depicted in blue, and the right semicircle is in red to illustrate the trajectory. This modification highlights the path traced by $\\exp(j\\theta)$, which follows a semicircular trajectory, as indicated by the blue and red segments for the appropriate values of $\\phi$. p.9 Eq. (25) $^$This is the most important revision that is needed$^$ (Later results will have to be modified accordingly). Let me explain: Equations (4) to (6) have been carefully devised so that the $\\theta$ domain is $(-\\infty,\\infty)$. Then Eq. (7) is the crucial coordinate definition. The $(-\\infty,\\infty)$ $\\theta$ domain is needed so that the product is a closed operation. However, in (25) what happens if $\\theta$ is larger than $\|\\frac{\\pi}{2}\|$? $\\exp(i\\phi)$ and $\\exp(j\\theta)$ are different functions, the ``Euler identity'' type expression for the latter involve your $\\mathrm{cos\\pi}(\\theta)$ and $\\mathrm{sin\\pi}(\\theta)$ functions. Response: Following the suggestion of reviewer, we have revised the text accordingly. In Equations (7) and (25), in general, \$\\phi \\in \\mathbb{R}\$, \$\\theta \\in \\mathbb{R}\$, \$e^{i\\phi}\$ is \$2\\pi\$-periodic and \$e^{j\\theta}\$ is \$\\pi\$-periodic. To normalize these angles, the \$2\\pi\$ modulo operation is applied to \$\\phi\$ to restrict it to the principal range \$[0, 2\\pi)\$. Similarly, the \$\\pi\$ modulo operation is used to adjust \$\\theta\$ to lie within the principal range \$[- \\pi/2, \\pi/2]\$. Specifically, if \$\\theta \\notin [-\\pi/2, \\pi/2]\$, it is normalized to this range by adding or subtracting a multiple of \$\\pi\$, i.e., by applying the transformation \$\\theta \\mapsto \\theta + n\\pi\$, where \$n \\in \\mathbb{Z}\$. Therefore, the azimuth angle $\\phi$ is considered $2\\pi$-periodic, and the elevation angle $\\theta$ is considered $\\pi$-periodic as shown in Fig.~4. p.9 Reconsider your assertion that ``This multiplication is continuous, meaning there are no discontinuities or jumps in the resulting sphere.'' Evaluate, for example the product $(e^{i\\phi_1}e^{j\\theta_1})\\times (e^{j\\theta_2})$, where $\\theta_1=0.9\\frac{\\pi}{2}$ and $\\theta_2=0.2\\frac{\\pi}{2}$. Also, reconsider Remark 2. Response: Based on the suggestion of the reviewer, we have removed the statement. We have also revised Remark 2 accordingly. p.9 In line Eq. after Eq. (26) implies $\\phi=0 \\mod \\pi$. Response: Thank you for suggestion, we have noted it. p.12 Remark 4 seems unnecessary and it has major problems. Notice that neutral element in the two operations is different. Response: Thank you for suggestion, we have removed it. p.12 Eq. (34), has to be revised. Remember that the series representation and the Euler identity are two different things. Since the domain of $\\mathrm{cos\\pi}(\\theta)$ and $\\mathrm{sin\\pi}(\\theta)$ is $(-\\infty,\\infty)$, problems arise since $e^{j\\theta}$ domain has been restricted to $\\pm \\frac{\\pi}{2}$ interval. But if the product is closed it has to be defined in $(-\\infty,\\infty)$, just a $\\mathrm{cos\\pi}(\\theta)$ and $\\mathrm{sin\\pi}(\\theta)$ are. It seems to me that you have to define the $e^{j\\theta}$ series using (4) to (6). Strictly speaking you should then label this `exponential' function differently, since it differs from the usual complex exponential. Response: Thank you for suggestion, now we have revised domain of $\\theta$ as $\\theta\\in\\mathbb{R}$ in (34). Moreover, in general, $\\phi \\in \\mathbb{R}$, $\\theta \\in \\mathbb{R}$, $e^{i\\phi}$ is $2\\pi$ periodic and $e^{j\\theta}$ is $\\pi$ periodic. Thus to normalize these angles, the $2\\pi$ modulo operation is applied to $\\phi$ to confine it within the principal range $[0, 2\\pi)$, and the $\\pi$ modulo operation is used to adjust $\\theta$ to lie within the principal range $[-\\pi/2, \\pi/2]$. This suggests that we can also consider $\\theta\\in[-\\pi/2,\\pi/2]$ in (34). p.14 Some of your assertions and/or proofs have to be carefully worked out. Take for example Proposition1, 8), let $g_1=e^{i\\frac{\\pi}{4}}e^{j\\frac{\\pi}{4}}\\ne 0$ and $g_2=e^{i\\frac{\\pi}{4}}e^{j\\frac{\\pi}{4}}\\ne 0$, what is the value of $g_1g_2=e^{i\\frac{\\pi}{2}}e^{j\\frac{\\pi}{2}}$? Response: There is a persistent nonuniqueness in representations that arises in specific scenario within the spherical coordinate system (SCS): $Z$-axis $(\\theta = \\pm \\pi/2)$: In this specific scenario where $\\theta$ equals $\\pi/2$ or $-\\pi/2$, for a fixed $r$, the azimuthal angle $\\phi$ can assume any value, as explicitly revealed by equation (2). Consequently, a vector situated along the $Z$-axis possesses an infinite array of representations as $(r, \\phi, \\pm \\pi/2)$. This implies that vectors characterized by diverse values of $r$, $\\phi$, and $\\theta$ can be aligned along the $Z$-axis by adjusting $\\theta$ to $\\pm \\pi/2$. Once aligned along the $Z$-axis, alterations in the polar angle $\\phi$ do not induce any alteration in the orientation of the vector. Thus, in these instances, manipulation of the values of one or more of the other coordinates can be undertaken without inducing displacement of the point in question. Therefore, if $g_1=e^{i\\frac{\\pi}{4}}e^{j\\frac{\\pi}{4}}\\ne 0$ and $g_2=e^{i\\frac{\\pi}{4}}e^{j\\frac{\\pi}{4}}\\ne 0$, then $g_1g_2=e^{i\\frac{\\pi}{2}}e^{j\\frac{\\pi}{2}}=j$. This is also same as $e^{i\\phi}e^{j\\frac{\\pi}{2}}=j$ for any $\\phi\\in[0,2\\pi)$, which is explained in Eqns. (27), (28) and (29). Moreover, the azimuth angle \$\\phi\$ in the expression \$e^{i\\phi}e^{j\\frac{\\pi}{2}}=j\$ indicates that any subsequent variation in the elevation angle \$\\theta\$ results in the displacement of the point from the \$Z\$-axis towards the direction specified by \$\\phi\$. Reviewer #2: Reviewer #2: In this paper the authors present a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Moreover, they elucidated the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Importantly, they explored its utility in point cloud image processing. The paper is well supported by the examples and figures which puts life to the paper. The paper is well written and fits to the scope of the journal. I recommend acceptance. Response: Authors would like to thank the reviewer for recommendation to accept the paper, and devoting time to provide detailed comments and observations. Thanks and Regards, Pushpendra Singh (On behalf of all authors) Attachments Attachment Submitted filename: PONE_RL_R1.pdf https://doi.org/10.1371/journal.pone.0312502.r002
22 Sep 2024 Decision Letter - Sania Asif, Editor PONE-D-24-24879R1On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplicationPLOS ONE Dear Dr. Singh, Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process. Please submit your revised manuscript by Nov 06 2024 11:59PM . If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file. Please include the following items when submitting your revised manuscript: A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'. An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'. If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter. If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols. We look forward to receiving your revised manuscript. Kind regards, Sania Asif Guest Editor PLOS ONE Journal Requirements: Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice. Additional Editor Comments: Dear Prof. Singh, Based on reviewer #1's report, I suggest authors to go through another round of minor revision and address the attached reviewer's remarks. Thank you Best Regards Sania Asif [Note: HTML markup is below. Please do not edit.] Reviewers' comments: Reviewer's Responses to Questions Comments to the Author 1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation. Reviewer #1: All comments have been addressed ******** 2. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #1: Yes ****** 3. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: Yes ****** 4. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #1: Yes ****** 5. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: Yes ****** 6. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #1: Please send attached comments to authors. A suggestion about how to deal with an exponential function appropriate for their purposes is outlined. ****** 7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #1: Yes ******** [NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.] While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. Attachments Attachment Submitted filename: hypercomplex2-plosone-rev2.pdf https://doi.org/10.1371/journal.pone.0312502.r003
Revision 2
4 Oct 2024 Author Response Response Letter PLOS ONE Dear Editor: First, please accept our thanks for devoting time and coordinating the reviews of the submitted manuscript Ref. No. PONE-D-24-24879 titled ``On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication" for possible publication in PLOS ONE. We submitted the manuscript on June 19, 2024, and received review comments from the reviewers on August 31, 2024. The second round of review comments was received on September 23, 2024. We sincerely appreciate the time and effort by the anonymous reviewers dedicated to evaluating our manuscript and offering valuable feedback, which has significantly contributed to enhancing its quality. We have done our best to address the reviewers' comments. We, hereby, submit: (1) a response letter with detailed replies to each of the reviewers' comments and (2) a revised manuscript that we believe to have significantly improved after incorporating the reviewers' suggestions. Please do thank the reviewers on our behalf for their time and effort. We also thank you for your work as the editor. We have also acknowledged editors and anonymous reviewers in the acknowledgments section of the revised manuscript. Thanks and regards, Authors Response to Reviewers: PONE-D-24-24879} Title: On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication Reviewer #1 Response: The authors thank the reviewer for providing detailed comments, pointing out errors, and helping us to improve the quality of the manuscript. The authors thank the reviewer for doing some of our homework. Based on the suggestions of the reviewers, we have incorporated all required changes into the revised manuscript. We have redefined the Equations (4), (5) and (6) as \\begin{align} \\tag{4} & \\mathrm{\\,cos\\pi}(\\theta)=\\begin{cases} \\cos(\\theta), & \\text{if } \\theta\\in \\bigcup\\limits_{n=-\\infty}^{\\infty}[-\\frac{\\pi}{2}+2\\pi n,\\frac{\\pi}{2}+2\\pi n],\\\\ -\\cos(\\theta), & \\text{if } \\theta\\in \\bigcup\\limits_{n=-\\infty}^{\\infty}(\\frac{\\pi}{2}+2\\pi n,\\frac{3\\pi}{2}+2\\pi n), \\end{cases} \\label{cospi1}\\\\ \\text{and } & \\mathrm{\\,sin\\pi}(\\theta)=\\begin{cases} \\sin(\\theta), & \\text{if } \\theta\\in \\bigcup\\limits_{n=-\\infty}^{\\infty}[-\\frac{\\pi}{2}+2\\pi n,\\frac{\\pi}{2}+2\\pi n],\\\\ -\\sin(\\theta), & \\text{if } \\theta\\in \\bigcup\\limits_{n=-\\infty}^{\\infty}(\\frac{\\pi}{2}+2\\pi n,\\frac{3\\pi}{2}+2\\pi n), \\end{cases} \\tag{5} \\label{sinpi1} \\end{align} where the intervals form a complete, non-overlapping partition of the real line as \\begin{align} \\tag{6} \\mathbb{R} = \\left\\{\\bigcup\\limits_{n=-\\infty}^{\\infty} \\left[ -\\frac{\\pi}{2} + 2n\\pi, \\frac{\\pi}{2} + 2n\\pi \\right]\\right\\} \\bigcup \\left\\{\\bigcup\\limits_{n=-\\infty}^{\\infty} \\left( \\frac{\\pi}{2} + 2n\\pi, \\frac{3\\pi}{2} + 2n\\pi \\right)\\right\\}. \\end{align} %item end For the elevation angles, we employ the transformation for all $\\theta\\in\\mathbb{R}$ as follows: \\begin{align} \\tag{26} \\theta \\mapsto \\theta\\, \\mathrm{m\\hat{o}d}\\, \\pi= \\begin{cases} \\theta & \\text{if } \\theta\\in [-\\pi/2,\\pi/2],\\\\ \\theta - m\\pi & \\text{if } \\theta > \\frac{\\pi}{2}, \\\\ \\theta + m\\pi & \\text{if } \\theta < -\\frac{\\pi}{2}, \\end{cases} \\label{modpi} \\end{align} where \$ m \$ is the smallest positive integer such that the result lies within the interval \$ \\left[ -\\frac{\\pi}{2}, \\frac{\\pi}{2} \\right] \$. The standard $\\pi$ modulo function, $\\theta\\mod \\pi$, produces results in the interval $[0, \\pi)$ that can be further adjusted to $[-\\pi/2, \\pi/2)$ or $(-\\pi/2, \\pi/2]$. Thus, $\\theta\\, \\mathrm{m\\hat{o}d}\\, \\pi= \\theta \\mod \\pi$, almost everywhere. Based on the suggestions of the reviewer and above equations, we have revised Result 1 to enhance clarity and eliminate any potential contradictions or confusion. We have also added the following relations after Equation (36) \\begin{align} & \\mathrm{\\,cos\\pi}(\\theta)=\\begin{cases} \\mathrm{Re}\\left(e^{j\\theta}\\right)=\\sum\\limits_{n=0}^\\infty(-1)^{n} \\frac{(\\theta)^{2n}}{(2n)!}, & \\text{if } \\theta\\in \\bigcup\\limits_{n=-\\infty}^{\\infty}[-\\frac{\\pi}{2}+2\\pi n,\\frac{\\pi}{2}+2\\pi n],\\\\ \\mathrm{Re}\\left(-e^{j\\theta}\\right)=-\\sum\\limits_{n=0}^\\infty(-1)^{n} \\frac{(\\theta)^{2n}}{(2n)!}, & \\text{if } \\theta\\in \\bigcup\\limits_{n=-\\infty}^{\\infty}(\\frac{\\pi}{2}+2\\pi n,\\frac{3\\pi}{2}+2\\pi n), \\end{cases} \\nonumber\\\\ \\text{and } & \\mathrm{\\,sin\\pi}(\\theta)=\\begin{cases} \\mathrm{Im}\\left(e^{j\\theta}\\right)=\\sum\\limits_{n=0}^\\infty(-1)^{n} \\frac{(\\theta)^{2n+1}}{(2n+1)!}, & \\text{if } \\theta\\in \\bigcup\\limits_{n=-\\infty}^{\\infty}[-\\frac{\\pi}{2}+2\\pi n,\\frac{\\pi}{2}+2\\pi n],\\\\ \\mathrm{Im}\\left(-e^{j\\theta}\\right)=-\\sum\\limits_{n=0}^\\infty(-1)^{n} \\frac{(\\theta)^{2n+1}}{(2n+1)!}, & \\text{if } \\theta\\in \\bigcup\\limits_{n=-\\infty}^{\\infty}(\\frac{\\pi}{2}+2\\pi n,\\frac{3\\pi}{2}+2\\pi n). \\end{cases} \\nonumber \\end{align} Thanks and Regards, Pushpendra Singh (On behalf of all authors) Attachments Attachment Submitted filename: PONE_R2_RL.pdf https://doi.org/10.1371/journal.pone.0312502.r004
8 Oct 2024 Decision Letter - Sania Asif, Editor On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication PONE-D-24-24879R2 Dear Dr. Singh, We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements. Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication. An invoice will be generated when your article is formally accepted. Please note, if your institution has a publishing partnership with PLOS and your article meets the relevant criteria, all or part of your publication costs will be covered. Please make sure your user information is up-to-date by logging into Editorial Manager at Editorial Manager® and clicking the ‘Update My Information' link at the top of the page. If you have any questions relating to publication charges, please contact our Author Billing department directly at authorbilling@plos.org. If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. Kind regards, Sania Asif Guest Editor PLOS ONE Additional Editor Comments (optional): Dear Professor Singh, It's my pleasure to inform you that your paper entitled "On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication" (PONE-D-24-24879R2) has been accepted by the editorial board of PLOS ONE. Thanks for your contribution to the Journal. Best Regards, Sania Asif Reviewers' comments: Reviewer's Responses to Questions Comments to the Author Reviewer #1: All comments have been addressed ******** 2. Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented. Reviewer #1: (No Response) ****** 3. Has the statistical analysis been performed appropriately and rigorously? Reviewer #1: (No Response) ****** 4. Have the authors made all data underlying the findings in their manuscript fully available? The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified. Reviewer #1: (No Response) ****** 5. Is the manuscript presented in an intelligible fashion and written in standard English? PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here. Reviewer #1: (No Response) ****** 6. Review Comments to the Author Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters) Reviewer #1: (No Response) ****** 7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files. If you choose “no”, your identity will remain anonymous but your review may still be made public. Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. Reviewer #1: No ******** https://doi.org/10.1371/journal.pone.0312502.r005
Formally Accepted
14 Oct 2024 Acceptance Letter - Sania Asif, Editor PONE-D-24-24879R2 PLOS ONE Dear Dr. Singh, I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now being handed over to our production team. At this stage, our production department will prepare your paper for publication. This includes ensuring the following: * All references, tables, and figures are properly cited * All relevant supporting information is included in the manuscript submission, * There are no issues that prevent the paper from being properly typeset If revisions are needed, the production department will contact you directly to resolve them. If no revisions are needed, you will receive an email when the publication date has been set. At this time, we do not offer pre-publication proofs to authors during production of the accepted work. Please keep in mind that we are working through a large volume of accepted articles, so please give us a few weeks to review your paper and let you know the next and final steps. Lastly, if your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org. If we can help with anything else, please email us at customercare@plos.org. Thank you for submitting your work to PLOS ONE and supporting open access. Kind regards, PLOS ONE Editorial Office Staff on behalf of Dr. Sania Asif Guest Editor PLOS ONE https://doi.org/10.1371/journal.pone.0312502.r006

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