Raylu

raylu

Ishan Pandey: Hi Ray, welcome to our series “Behind the Startup.” Please tell us about yourself and the story behind ? Bit.Country I started entrepreneurship in 2012; the first startup was , which helps people kick-start their tech career by solving the problem that recent graduates lack industrial experience. Currently, Industry Connect has operations in most English-speaking countries, approved by SG IMDA. Having graduates working for more than 500 companies. Ray Lu: IndustryConnect.org With the top students that Ray Lu selected, they also work together at MVP STUDIO, a startup incubator. Over the years, LG Trust Real estate (BI enabled property investment company), , and the Bit.Country / Metaverse.Network. GlobalTalent.io The culture is a journey together. Business is created based on accelerating individual growth by providing a lifetime roadmap, for example, top students, employees, managerial roles, and co-founders and CEO of their own startups.  Bit.Country & Metaverse. The network was also founded the same way with three other co-founders who were students at many years ago. IndustryConnect.org Ishan Pandey: What are your views on the metaverse? Further, what future do you see in the next 2 years for the metaverse industry? Since we registered Bit.country on Jan 22nd, 2018, we envisaged that the blockchain-backed virtual world (now metaverse) would become a medium for crypto to be used for mainstream use. There will be new native supply and demand in the metaverse and creating new career opportunities. Ray Lu: Humans have never stopped expanding their consciousness and sphere of exploration. We are composing a new chapter in the web3.0 paradigm for mass adoption with sustainable technologies. Ever since the internet came into our lives, we have benefited from the high speed of information exchange, and then blockchain networks provided us with a trustless flow of value. We now envisage that the metaverse will bring civilization a highway of imagination, happiness, and opportunities. Ishan Pandey: Please tell us a bit of Bit.country and how users can create their metaverse? Everyone can start their metaverse for their people with the 3D world, NFTs, play-to-earn & build communities to earn, and takes community engagement to a new dimension on web3.0. Ray Lu: With at least one land block, users can launch a metaverse with a map engine, 3D world, local governance, and marketplace.  Metaverse owner can invite their community to check out their design and culture. The land block is subdivided into land units or estates, and users can buy the land units and become a stakeholder in the metaverse. User with the land has mining power to earn in-game current BIT used as base material or energy source. Users can join various earning opportunities in the community, such as participating in tasks or games created by users. According to you, what are some of the primary ways metaverse will alter users’ lives? Ishan Pandey: Ray Lu: •	Connect with people in a metaverse native way. •	A new identity with new personal traits reflected in their avatar, e.g. MetaSoul •	Earn through new supply and demand native to the metaverse •	The community can be perpetuated and governed by stakeholders that can last as long as bitcoin exists. Ishan Pandey: Researchers are now concerned about the human mind’s ability to process competing binaries while maintaining equilibrium in the natural and digital world. Is the metaverse weakening our ties to the real world? Humans have never stopped expanding their consciousness and sphere of exploration. We are composing a new chapter in the web3.0 paradigm with sustainable technologies. To a certain extent, the digital world brings efficiency and less environmentally impactful activities. Nowadays, the digital world has already become the main medium for people to learn, connect and entertain. Ray Lu: Metaverse’s role will bring the current web2.0 experience to a new level with open economies and governance. Just as what web2.0 had brought to the world, we believe web3/metaverse will create exponential opportunities for people for better living. Education has always been our passion. We are also launching an initiative through , the global tech career academy, to bring education efficiency into the metaverse and web3.0. IndustryConnect.org Is the metaverse weakening our ties to the real world? It may or may not. We see it as an extension to the real world and potentially make it better. Ishan Pandey: Some argue that the metaverse is little more than marketing hype, while others argue it will revolutionize civilization. According to you, what are the major hurdles in the industry which may impact the Metaverse ecosystem? Our view is metaverse will bring our civilization to a new level. While blockchain technology unlocks the trustless flow of value, a medium built on the blockchain is the new place for opportunities regardless of their location and nationalities. A private key is a key to a career and the door to many opportunities. Ray Lu: Protocols used in the metaverse will act as the business rules and it does not care who you are and where you are from. It is a fair system to give the same opportunities to everyone willing to learn and put the effort in. For example, the rich will create new demand in the metaverse, such as growing their influence, and metaverse will unlock more opportunities for the new demand and make them available to those who need it regardless of where they are. Ishan Pandey: Brands have developed arenas, services, and branded characters inside games as some of the more exciting Metaverse deployments. According to you, will we soon witness mass shifts across all sectors to this digital world? Further, why are brands building in the metaverse? Brand accessibility in metaverse can be much easier than in the real world. That is why Brands are now considering building up the presence and providing services in the metaverse. This has happened already on web2.0 e.g. websites, social channels. If metaverse brings the next level of user experience, the trend is inevitable. Ray Lu: We are disrupting the current creators’ value system built by web2.0, as traditional social platforms are harvesting the value from the community’s time, only distributing a portion of the profits to the creators. At the same time, members are only being entertained but not rewarded. We are building a future for metaverse owners to have the ability to grow exponentially while letting the members take ownership and earn through a variety of opportunities, benefitting directly from the development of the metaverses they are in. We see community owners and big brands on web2.0 have a huge desire to build up their metaverse. Ishan Pandey: What role has the pandemic and the persistent lockdown across the globe affected in the development of the blockchain industry and what will the scenario look like in the post-pandemic era? Pandemic forced people to stay in the same place, and some people had lost their jobs. Metaverse and blockchain technologies had proven for bringing opportunities during the lockdown. From NFT sales to Play2earn protocols, it provided income and brought meals to the families. Ray Lu: In the post-pandemic era, we see the trend will continue, especially some of the existing business models are being shifted during the pandemic, people’s view has changed, they will look for a sustainable solution to operating their business and constantly seek for new opportunities to create value. We believe metaverse/web3 is the solution for the needs. Ishan Pandey: According to you, what new trends are we going to see in the blockchain industry? According to the decentralized nature of blockchain, there will be more and more community-driven business models to come into our life. We envisaged that utility tokens, especially NFTs would be more utilized in the real world. And metaverse is the native place to have these created in the first place before it comes to the real world. Ray Lu: Disclaimer: The purpose of this article is to remove informational asymmetry existing today in our digital markets by performing due diligence, asking the right questions, and equipping readers with better opinions to make informed decisions.

The Metaverse Will Bring Our Civilization to a New Level

Advancing the Detection of Ultralight Vector Dark Matter through Novel Analytical Strategies

Authors:
(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;
(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;
(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;
(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix D The Case of the Gradient of a Scalar 3 Statistical Analysis We now consider the projected exclusion limits that a generic experiment would be able to set using our three-peak analysis. To do this, we use a series of likelihood-ratio tests. 3.1 Signal Likelihood For our likelihood, we follow a hybrid frequentist-Bayesian approach, defining a marginalized likelihood in which all nuisance parameters are integrated out. In our case, these are the random Rayleigh parameters, α, and random uniform DM phases, φ. Such a hybrid approach has already been used in the context of ultralight bosonic dark matter [42, 79]. Our work differs from Ref. [79] since they focused on an axion-like signal as opposed to that from vector DM. It goes beyond Ref. [42] since they did not consider the peaks arising from the rotation of the Earth in their analysis. The full likelihood over all frequency space is then given by the product of the likelihoods in each frequency bin, where pi represents the excess power density in the ith frequency bin, p is the full data vector, and the product runs over all Nbins frequency bins. Ultimately, since our signal only manifests in three bins, it suffices for us to consider only those bins that could potentially contain a signal, and we may ignore all other bins. We can express the likelihood in this way because each bin is statistically uncorrelated, as we show in Appendix C. This is in contrast with the analysis performed in Ref. [45], where a similar study was conducted in the case of the gradient of a scalar in the time domain. There, a complicated covariance matrix had to be computed to account for correlations in the signal at different times. In Fourier space, these covariances disappear. The power of performing this analysis in the frequency domain is thus not only that the signal is contained within a small number of bins, but also that these bins are statistically independent, which allows us to treat the statistics in a significantly simpler way. Crucially, once the latitude of the experiment, ϕ, is fixed, the likelihood depends on the product of all experimental variables via the dimensionless parameter β. This means that we can set a more holistic limit that is independent of the specifics of an experiment. Once the form of A (which depends on both the experiment and the DM model), the observation time Tobs, and the noise profile σ are known, the ensuing limit on β can be recast to one on the model parameters of interest. This makes our analysis, both the results and overall logic, as generally useful as possible. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Authors: Authors: (1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work; (2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work; (3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University; (4) Christopher Tunnell, Department of Physics and Astronomy, Rice University. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Calculating the Stochastic Wave Vector Dark Matter Signal 2.1 The Dark Photon Field 2.1 The Dark Photon Field 2.2 The Detector Signal 2.2 The Detector Signal 3 Statistical Analysis and 3.1 Signal Likelihood 3 Statistical Analysis and 3.1 Signal Likelihood 3.2 Projected Exclusions 3.2 Projected Exclusions 4 Application to Accelerometer Studies 4 Application to Accelerometer Studies 4.1 Recasting Generalised Limits onto B − L Dark Matter 4.1 Recasting Generalised Limits onto B − L Dark Matter 5 Future Directions 5 Future Directions 6 Conclusions, Acknowledgments, and References 6 Conclusions, Acknowledgments, and References A Equipartition between Longitudinal and Transverse Modes A Equipartition between Longitudinal and Transverse Modes B Derivation of Marginal Likelihood with Stochastic Field Amplitude B Derivation of Marginal Likelihood with Stochastic Field Amplitude C Covariance Matrix C Covariance Matrix D The Case of the Gradient of a Scalar D The Case of the Gradient of a Scalar 3 Statistical Analysis We now consider the projected exclusion limits that a generic experiment would be able to set using our three-peak analysis. To do this, we use a series of likelihood-ratio tests. 3.1 Signal Likelihood For our likelihood, we follow a hybrid frequentist-Bayesian approach, defining a marginalized likelihood in which all nuisance parameters are integrated out. In our case, these are the random Rayleigh parameters, α, and random uniform DM phases, φ. Such a hybrid approach has already been used in the context of ultralight bosonic dark matter [42, 79]. Our work differs from Ref. [79] since they focused on an axion-like signal as opposed to that from vector DM. It goes beyond Ref. [42] since they did not consider the peaks arising from the rotation of the Earth in their analysis. The full likelihood over all frequency space is then given by the product of the likelihoods in each frequency bin, where pi represents the excess power density in the i th frequency bin, p is the full data vector, and the product runs over all Nbins frequency bins. Ultimately, since our signal only manifests in three bins, it suffices for us to consider only those bins that could potentially contain a signal, and we may ignore all other bins. We can express the likelihood in this way because each bin is statistically uncorrelated, as we show in Appendix C. This is in contrast with the analysis performed in Ref. [45], where a similar study was conducted in the case of the gradient of a scalar in the time domain. There, a complicated covariance matrix had to be computed to account for correlations in the signal at different times. In Fourier space, these covariances disappear. The power of performing this analysis in the frequency domain is thus not only that the signal is contained within a small number of bins, but also that these bins are statistically independent, which allows us to treat the statistics in a significantly simpler way. th Crucially, once the latitude of the experiment, ϕ, is fixed, the likelihood depends on the product of all experimental variables via the dimensionless parameter β. This means that we can set a more holistic limit that is independent of the specifics of an experiment. Once the form of A (which depends on both the experiment and the DM model), the observation time Tobs, and the noise profile σ are known, the ensuing limit on β can be recast to one on the model parameters of interest. This makes our analysis, both the results and overall logic, as generally useful as possible. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Using Three-Peak Analysis for Projected Exclusion Limits in Vector Dark Matter Detection

Authors:
(1) Halil Yigit Oksuz, Control Systems Group at Technische Universitat Berlin, Germany and Exzellenzcluster Science of Intelligence, Technische Universitat Berlin, Marchstr. 23, 10587, Berlin, Germany;
(2) Fabio Molinari, Control Systems Group at Technische Universitat Berlin, Germany;
(3) Henning Sprekeler, Exzellenzcluster Science of Intelligence, Technische Universit¨at Berlin, Marchstr. 23, 10587, Berlin, Germany and Modelling Cognitive Processes Group at Technische Universit¨at Berlin, Germany;
(4) Jorg Raisch, Control Systems Group at Technische Universitat Berlin, Germany and Exzellenzcluster Science of Intelligence, Technische Universitat Berlin, Marchstr. 23, 10587, Berlin, Germany. Table of Links Abstract and Introduction Problem Setup Federated fair over-the-air learning (FedAir) Algorithm Convergence Properties Numerical Example Conclusion and References VI. CONCLUSION In this paper, we have introduced the FedFAir algorithm which uses Over-the-Air Computation to carry out efficient decentralized learning while providing fairness and improved performance. We have shown that the FedFAir algorithm converges to an optimal solution of the minimax problem. Furthermore, we have also illustrated our theoretical findings with a numerical example. Future research will include the development of resilient federated learning algorithms when there are malicious agents in the system. REFERENCES [1] B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas, “Communication-efficient learning of deep networks from decentralized data,” in Artificial intelligence and statistics. PMLR, 2017, pp. 1273–1282. [2] V. Smith, C.-K. Chiang, M. Sanjabi, and A. S. Talwalkar, “Federated multi-task learning,” Advances in neural information processing systems, vol. 30, 2017. [3] K. Bonawitz, H. Eichner, W. Grieskamp, D. Huba, A. Ingerman, V. Ivanov, C. Kiddon, J. Konecny, S. Mazzocchi, B. McMahan et al., “Towards federated learning at scale: System design,” Proceedings of Machine Learning and Systems, vol. 1, pp. 374–388, 2019. [4] Q. Yang, Y. Liu, T. Chen, and Y. Tong, “Federated machine learning: Concept and applications,” ACM Transactions on Intelligent Systems and Technology (TIST), vol. 10, no. 2, pp. 1–19, 2019. [5] A. Nedic, “Distributed gradient methods for convex machine learning problems in networks: Distributed optimization,” IEEE Signal Processing Magazine, vol. 37, no. 3, pp. 92–101, 2020. [6] V. P. Chellapandi, A. Upadhyay, A. Hashemi, and S. H. Zak, “On ˙ the convergence of decentralized federated learning under imperfect information sharing,” IEEE Control Systems Letters, 2023. [7] T. Omori and K. Kashima, “Combinatorial optimization approach to client scheduling for federated learning,” IEEE Control Systems Letters, 2023. [8] T. Sery, N. Shlezinger, K. Cohen, and Y. C. Eldar, “Over-the-air federated learning from heterogeneous data,” IEEE Transactions on Signal Processing, vol. 69, pp. 3796–3811, 2021. [9] T. Gafni, N. Shlezinger, K. Cohen, Y. C. Eldar, and H. V. Poor, “Federated learning: A signal processing perspective,” IEEE Signal Processing Magazine, vol. 39, no. 3, pp. 14–41, 2022. [10] M. Ye, X. Fang, B. Du, P. C. Yuen, and D. Tao, “Heterogeneous federated learning: State-of-the-art and research challenges,” ACM Computing Surveys, vol. 56, no. 3, pp. 1–44, 2023. [11] J. Koneˇcn `y, H. B. McMahan, F. X. Yu, P. Richt´arik, A. T. Suresh, and D. Bacon, “Federated learning: Strategies for improving communication efficiency,” arXiv preprint arXiv:1610.05492, 2016. [12] T. Li, A. K. Sahu, A. Talwalkar, and V. Smith, “Federated learning: Challenges, methods, and future directions,” IEEE Signal Processing Magazine, vol. 37, no. 3, pp. 50–60, 2020. [13] P. Kairouz, H. B. McMahan, B. Avent, A. Bellet, M. Bennis, A. N. Bhagoji, K. Bonawitz, Z. Charles, G. Cormode, R. Cummings et al., “Advances and open problems in federated learning,” Foundations and Trends® in Machine Learning, vol. 14, no. 1–2, pp. 1–210, 2021. [14] H. Y. Oksuz, F. Molinari, H. Sprekeler, and J. Raisch, “Federated learning in wireless networks via over-the-air computations,” in 2023 62nd IEEE Conference on Decision and Control (CDC), 2023, pp. 4379–4386. [15] F. Molinari, N. Agrawal, S. Sta´nczak, and J. Raisch, “Max-consensus over fading wireless channels,” IEEE Transactions on Control of Network Systems, vol. 8, no. 2, pp. 791–802, 2021. [16] M. Frey, I. Bjelakovi´c, and S. Sta´nczak, “Over-the-air computation in correlated channels,” IEEE Transactions on Signal Processing, vol. 69, pp. 5739–5755, 2021. [17] K. Yang, T. Jiang, Y. Shi, and Z. Ding, “Federated learning via overthe-air computation,” IEEE Transactions on Wireless Communications, vol. 19, no. 3, pp. 2022–2035, 2020. [18] F. Molinari and J. Raisch, “Exploiting wireless interference for distributively solving linear equations,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 2999–3006, 2020. [19] D. Bertsekas, A. Nedic, and A. Ozdaglar, Convex analysis and optimization. Athena Scientific, 2003, vol. 1. [20] M. Mohri, G. Sivek, and A. T. Suresh, “Agnostic federated learning,” in International Conference on Machine Learning. PMLR, 2019, pp. 4615–4625. [21] Z. Hu, K. Shaloudegi, G. Zhang, and Y. Yu, “Federated learning meets multi-objective optimization,” IEEE Transactions on Network Science and Engineering, vol. 9, no. 4, pp. 2039–2051, 2022. [22] D. P. Bertsekas, “Necessary and sufficient conditions for a penalty method to be exact,” Mathematical programming, vol. 9, no. 1, pp. 87–99, 1975. [23] K. Srivastava, A. Nedi´c, and D. Stipanovi´c, “Distributed min-max optimization in networks,” in 2011 17th International Conference on Digital Signal Processing (DSP). IEEE, 2011, pp. 1–8. [24] R. Ahlswede, “Multi-way communication channels,” in Second International Symposium on Information Theory: Tsahkadsor, Armenia, USSR, Sept. 2-8, 1971, 1973. [25] A. Giridhar and P. Kumar, “Toward a theory of in-network computation in wireless sensor networks,” IEEE Communications Magazine, vol. 44, no. 4, pp. 98–107, apr 2006. [26] F. Molinari, N. Agrawal, S. Sta´nczak, and J. Raisch, “Over-the-air max-consensus in clustered networks adopting half-duplex communication technology,” IEEE Transactions on Control of Network Systems, 2022. [27] W. Rudin et al., Principles of mathematical analysis. McGraw-hill New York, 1976, vol. 3. [28] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2012. [29] A. F. Molisch, Wireless communications. John Wiley & Sons, 2012. [30] B. Sklar, “Rayleigh fading channels in mobile digital communication systems. i. characterization,” IEEE Communications magazine, vol. 35, no. 7, pp. 90–100, 1997. [31] R. G. Bartle and D. R. Sherbert, Introduction to real analysis. Wiley New York, 2000. [32] B. T. Polyak, Introduction to optimization. optimization software. Inc., Publications Division, New York, 1987. [33] Y. I. Alber, A. N. Iusem, and M. V. Solodov, “On the projected subgradient method for nonsmooth convex optimization in a hilbert space,” Mathematical Programming, vol. 81, pp. 23–35, 1998. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Halil Yigit Oksuz, Control Systems Group at Technische Universitat Berlin, Germany and Exzellenzcluster Science of Intelligence, Technische Universitat Berlin, Marchstr. 23, 10587, Berlin, Germany; (2) Fabio Molinari, Control Systems Group at Technische Universitat Berlin, Germany; (3) Henning Sprekeler, Exzellenzcluster Science of Intelligence, Technische Universit¨at Berlin, Marchstr. 23, 10587, Berlin, Germany and Modelling Cognitive Processes Group at Technische Universit¨at Berlin, Germany; (4) Jorg Raisch, Control Systems Group at Technische Universitat Berlin, Germany and Exzellenzcluster Science of Intelligence, Technische Universitat Berlin, Marchstr. 23, 10587, Berlin, Germany. Authors: Authors: (1) Halil Yigit Oksuz, Control Systems Group at Technische Universitat Berlin, Germany and Exzellenzcluster Science of Intelligence, Technische Universitat Berlin, Marchstr. 23, 10587, Berlin, Germany; (2) Fabio Molinari, Control Systems Group at Technische Universitat Berlin, Germany; (3) Henning Sprekeler, Exzellenzcluster Science of Intelligence, Technische Universit¨at Berlin, Marchstr. 23, 10587, Berlin, Germany and Modelling Cognitive Processes Group at Technische Universit¨at Berlin, Germany; (4) Jorg Raisch, Control Systems Group at Technische Universitat Berlin, Germany and Exzellenzcluster Science of Intelligence, Technische Universitat Berlin, Marchstr. 23, 10587, Berlin, Germany. Table of Links Abstract and Introduction Abstract and Introduction Problem Setup Problem Setup Federated fair over-the-air learning (FedAir) Algorithm Federated fair over-the-air learning (FedAir) Algorithm Convergence Properties Convergence Properties Numerical Example Numerical Example Conclusion and References Conclusion and References VI. CONCLUSION In this paper, we have introduced the FedFAir algorithm which uses Over-the-Air Computation to carry out efficient decentralized learning while providing fairness and improved performance. We have shown that the FedFAir algorithm converges to an optimal solution of the minimax problem. Furthermore, we have also illustrated our theoretical findings with a numerical example. Future research will include the development of resilient federated learning algorithms when there are malicious agents in the system. REFERENCES [1] B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas, “Communication-efficient learning of deep networks from decentralized data,” in Artificial intelligence and statistics. PMLR, 2017, pp. 1273–1282. [2] V. Smith, C.-K. Chiang, M. Sanjabi, and A. S. Talwalkar, “Federated multi-task learning,” Advances in neural information processing systems, vol. 30, 2017. [3] K. Bonawitz, H. Eichner, W. Grieskamp, D. Huba, A. Ingerman, V. Ivanov, C. Kiddon, J. Konecny, S. Mazzocchi, B. McMahan et al., “Towards federated learning at scale: System design,” Proceedings of Machine Learning and Systems, vol. 1, pp. 374–388, 2019. [4] Q. Yang, Y. Liu, T. Chen, and Y. Tong, “Federated machine learning: Concept and applications,” ACM Transactions on Intelligent Systems and Technology (TIST), vol. 10, no. 2, pp. 1–19, 2019. [5] A. Nedic, “Distributed gradient methods for convex machine learning problems in networks: Distributed optimization,” IEEE Signal Processing Magazine, vol. 37, no. 3, pp. 92–101, 2020. [6] V. P. Chellapandi, A. Upadhyay, A. Hashemi, and S. H. Zak, “On ˙ the convergence of decentralized federated learning under imperfect information sharing,” IEEE Control Systems Letters, 2023. [7] T. Omori and K. Kashima, “Combinatorial optimization approach to client scheduling for federated learning,” IEEE Control Systems Letters, 2023. [8] T. Sery, N. Shlezinger, K. Cohen, and Y. C. Eldar, “Over-the-air federated learning from heterogeneous data,” IEEE Transactions on Signal Processing, vol. 69, pp. 3796–3811, 2021. [9] T. Gafni, N. Shlezinger, K. Cohen, Y. C. Eldar, and H. V. Poor, “Federated learning: A signal processing perspective,” IEEE Signal Processing Magazine, vol. 39, no. 3, pp. 14–41, 2022. [10] M. Ye, X. Fang, B. Du, P. C. Yuen, and D. Tao, “Heterogeneous federated learning: State-of-the-art and research challenges,” ACM Computing Surveys, vol. 56, no. 3, pp. 1–44, 2023. [11] J. Koneˇcn `y, H. B. McMahan, F. X. Yu, P. Richt´arik, A. T. Suresh, and D. Bacon, “Federated learning: Strategies for improving communication efficiency,” arXiv preprint arXiv:1610.05492, 2016. [12] T. Li, A. K. Sahu, A. Talwalkar, and V. Smith, “Federated learning: Challenges, methods, and future directions,” IEEE Signal Processing Magazine, vol. 37, no. 3, pp. 50–60, 2020. [13] P. Kairouz, H. B. McMahan, B. Avent, A. Bellet, M. Bennis, A. N. Bhagoji, K. Bonawitz, Z. Charles, G. Cormode, R. Cummings et al., “Advances and open problems in federated learning,” Foundations and Trends® in Machine Learning, vol. 14, no. 1–2, pp. 1–210, 2021. [14] H. Y. Oksuz, F. Molinari, H. Sprekeler, and J. Raisch, “Federated learning in wireless networks via over-the-air computations,” in 2023 62nd IEEE Conference on Decision and Control (CDC), 2023, pp. 4379–4386. [15] F. Molinari, N. Agrawal, S. Sta´nczak, and J. Raisch, “Max-consensus over fading wireless channels,” IEEE Transactions on Control of Network Systems, vol. 8, no. 2, pp. 791–802, 2021. [16] M. Frey, I. Bjelakovi´c, and S. Sta´nczak, “Over-the-air computation in correlated channels,” IEEE Transactions on Signal Processing, vol. 69, pp. 5739–5755, 2021. [17] K. Yang, T. Jiang, Y. Shi, and Z. Ding, “Federated learning via overthe-air computation,” IEEE Transactions on Wireless Communications, vol. 19, no. 3, pp. 2022–2035, 2020. [18] F. Molinari and J. Raisch, “Exploiting wireless interference for distributively solving linear equations,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 2999–3006, 2020. [19] D. Bertsekas, A. Nedic, and A. Ozdaglar, Convex analysis and optimization. Athena Scientific, 2003, vol. 1. [20] M. Mohri, G. Sivek, and A. T. Suresh, “Agnostic federated learning,” in International Conference on Machine Learning. PMLR, 2019, pp. 4615–4625. [21] Z. Hu, K. Shaloudegi, G. Zhang, and Y. Yu, “Federated learning meets multi-objective optimization,” IEEE Transactions on Network Science and Engineering, vol. 9, no. 4, pp. 2039–2051, 2022. [22] D. P. Bertsekas, “Necessary and sufficient conditions for a penalty method to be exact,” Mathematical programming, vol. 9, no. 1, pp. 87–99, 1975. [23] K. Srivastava, A. Nedi´c, and D. Stipanovi´c, “Distributed min-max optimization in networks,” in 2011 17th International Conference on Digital Signal Processing (DSP). IEEE, 2011, pp. 1–8. [24] R. Ahlswede, “Multi-way communication channels,” in Second International Symposium on Information Theory: Tsahkadsor, Armenia, USSR, Sept. 2-8, 1971, 1973. [25] A. Giridhar and P. Kumar, “Toward a theory of in-network computation in wireless sensor networks,” IEEE Communications Magazine, vol. 44, no. 4, pp. 98–107, apr 2006. [26] F. Molinari, N. Agrawal, S. Sta´nczak, and J. Raisch, “Over-the-air max-consensus in clustered networks adopting half-duplex communication technology,” IEEE Transactions on Control of Network Systems, 2022. [27] W. Rudin et al., Principles of mathematical analysis. McGraw-hill New York, 1976, vol. 3. [28] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2012. [29] A. F. Molisch, Wireless communications. John Wiley & Sons, 2012. [30] B. Sklar, “Rayleigh fading channels in mobile digital communication systems. i. characterization,” IEEE Communications magazine, vol. 35, no. 7, pp. 90–100, 1997. [31] R. G. Bartle and D. R. Sherbert, Introduction to real analysis. Wiley New York, 2000. [32] B. T. Polyak, Introduction to optimization. optimization software. Inc., Publications Division, New York, 1987. [33] Y. I. Alber, A. N. Iusem, and M. V. Solodov, “On the projected subgradient method for nonsmooth convex optimization in a hilbert space,” Mathematical Programming, vol. 81, pp. 23–35, 1998. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Boosting Fairness and Robustness in Over-the-Air Federated Learning: Conclusion and References

Author:
(1) K. Razi Naqvi, Department of Physics, Norwegian University of Science and Technology (NTNU), 7094 Trondheim, Norway Table of Links Abstract and 1 Introduction 2 Preliminary material 3 Comments elicited by the solution to Milne’s problem for B-particles 4 Flux to a spherical trap 5 Molecular motion in liquids 6 Concluding remarks and References 6 Concluding remarks The claim—motivated by the search for a better boundary condition—that the Lorentz model is a useful tool for investigating bimolecular reactions in solutions became credible only after the publication of Burschka and Titulaer’s numerical solution of the one-dimensional KKE [4]. Immediately prior to that, one critic expressed the “community opinion” by stating in a referee report (on an article co-authored by me) that the LBE “is absolutely useless in dealing with transport in liquids”, and insisted that this task is best handled by solving the KKE. Only then did the need arise for comparing the length scales of inverse and regular Brownian motion. The Trondheim group has shown that inverse Brownian motion, regular Brownian motion and the BGK-model are indistinguishable at the DE-level, provided that one uses the appropriate BC [31], namely that stated in Eq. (20). A Procrustean random walk model that allows no distribution of path lengths seems (to me) unphysical, much like the lattice model that informed Smoluchowksi’s thinking (about the boundary condition at an absorbing surface), and has misinformed generations of students as well as aficionados of chemical kinetics. Infinitely heavy B-particles, infinitely light L-particles, infinitely inflexible (about the constancy of their pathlenghts) R-particles are all fictions, but some fictions are more fruitful than others, and some are outright useless. Whether the fiction of R-particles will bear fruit (in the setting of diffusion-mediated reactions) or serve as a mere distraction remains to be seen. A Milne’s problem: calculating the density profiles The purpose of this appendix is to enable a reader of this article to generate the data used for plotting the density profiles shown in Fig. 1. The density data for L-particles were generated with the aid of a variational calculation [32], in which ℓ was used a the unit of length. The corresponding data for B-particles were computed by improving the results obtained by the Trondheim group through a half-range treatment [10], in which the 𝑁th order approximation for the particle density 𝑛 was expressed in the form and values of 𝑥0 (1.459877Λ), 𝑥𝑖 and 𝜆𝑖 (for 𝑖 =1–8) resulting from a ninth-order approximation (𝑁 = 8) were reported, and the values of 𝑛(𝑥) close to the wall were compared with those found by Marshall and Watson (M&W) on the basis of their exact analytical treatment [33]. The improvements consists of three minor changes: the value of 𝑥0 has been replaced by 𝑥0 = 1.460354Λ (the first seven figures of the exact result), one more term has been added, and the values of 𝑥𝑖 and 𝜆𝑖 for the last three terms (𝑖 =7–9) have been optimised in a least-squares fit to the numbers in column (A) of Table 1 of M&W. The complete set of {𝑥𝑖 , 𝜆𝑖} values (of mostly-analytical-partly-empirical origin) is displayed in Table 1, the upper part of which is identical with Table II of ref. [10]. For plotting the density of B-particles in Fig. 1, the length scale was changed from Λ to ℓ. References [1] M. von Smoluchowski. Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Physik. Zeit., 17:568–571 (1st instalment), 585–593 (2nd instalment), 1916. [2] M. von Smoluchowski. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z. Phys. Chem. (Leipzig), 92:129–168, 1917. [3] M. von Smoluchowski. Grundriß der Koagulationskinetik kolloider Lösungen. Kolloid Z., 21:98–104, 1917. [4] M. A. Burschka and U. M. Titulaer. The kinetic boundary layer for the Fokker-Planck equation with absorbing boundary. J. Stat. Phys., 25:569–582, 1981. [5] K. Razi Naqvi, K. J. Mork, and S. Waldenstrøm. Comment on “Steady, one-dimensional Brownian motion with an absorbing boundary”. J. Chem. Phys., 77(7):4263–4264, 1982. [6] K. Razi Naqvi, K. J. Mork, and S. Waldenstrøm. Reduction of the Fokker-Planck equation with an absorbing or reflecting boundary to the diffusion equation and the radiation boundary condition. Phys. Rev. Lett., 49:304–307, 1982. [7] K. Razi Naqvi, S. Waldenstrøm, and K. J. Mork. Kinetics of diffusion-mediated bimolecular reactions. A new theoretical framework. J. Phys. Chem., 86:4750–4756, 1982. [8] K. Razi Naqvi, S. Waldenstrøm, and K. J. Mork. On the rate of absorption of Brownian particles by a black sphere: The connection between the Fokker–Planck equation and the diffusion equation. J. Chem. Phys., 78(5):2710–2712, 1983. [9] Enrico Fermi (author) and Emilio Segre (chief editor). Collected Papers (Note E Memorie), (Vol. 1: Italy, 1921–1938). University of Chicago Press, Chicago, IL 1962. [10] K. Razi Naqvi, K. J. Mork, and S. Waldenstrm. Steady-state, one-dimensional Fokker-Planck equation with an absorbing boundary: A half-range treatment. Phys. Rev. A, 40:3405–3407, 1989. [11] S. Waldenstrøm, K. J. Mork, and K. Razi Naqvi. Milne problem for a hard-sphere Rayleigh gas: a study based on the Fokker-Planck equation. Phys. Rev. A, 28(3):1659–1661, 1983. [12] S. V. G. Menon, Vinod Kumar, and D. C. Sahni. Green’s function approach to the solution of the time dependent Fokker-Planck equation with an absorbing boundary. Physica A: Statistical Mechanics and its Applications, 135(1):63–79, 1986. [13] M. J. Lindenfeld and B. Shizgal. The Milne problem: A study of the mass dependence. Phys. Rev. A, 27:1657–1670, 1983. [14] Knud Andersen and Kurt E. Shuler. On the relaxation of the hard-sphere Rayleigh and Lorentz Gas. J. Chem. Phys., 40(3):633–650, 1964. [15] B. Davison. Neutron Transport Theory. OUP, Oxford, 1957. [16] Julian Keilson and James E. Storer. On Brownian motion, Boltzmann’s equation and the Fokker-Planck equation. Quarterly of Applied Mathematics, 10(3):243–253, 1952. [17] M. R. Hoare. The linear gas. In Ilya Prigogine and Stuart A. Rice, editors, Advance in Chemical Physics, volume 20, pages 135–214. Wiley-Interscience, New York, 1971. [18] Marie Smoluchowski. Sur le chemin moyen parcouru par les molécules d’un gaz et sur son rapport avec la théorie de la diffusion. Bull. Acad. Sci. Cracovie, pages 202–213, 1906. [19] K. Razi Naqvi. 100+ years of colossal confusion on colloidal coagulation. Part I: Smoluchowski’s work on absorbing boundaries. arXiv:2404.17021 [cond-mat.stat-mech], pages 1–20, 2024. [20] Robert M. Ziff. Flux to a trap. J. Stat. Phys., 65(5):1217–1233, 1991. [21] S. N. Majumdar, A. Comtet, and R. M. Ziff. Unified solution of the expected maximum of a discrete time random walk and the discrete flux to a spherical trap. J. Stat. Phys., 122(5):833– 856, 2006. [22] R. M. Ziff, S. N. Majumdar, and A. Comtet. General flux to a trap in one and three dimensions. J. Phys.: Condens. Matter, 19(6):065102, 2007. [23] H. L. Frisch and F. C. Collins Diffusional processes in the growth of aerosol particles. J. Chem. Phys., 20:1797–1803, 1952. [24] F. C. Collins and G. E. Kimball. Diffusion-controlled reactions in liquid solutions. Ind. Eng. Chem., 41(11):2551–2553, 1949. [25] D. C. Sahni. The effect of a black sphere on the flux distribution in an infinite moderator. J. Nucl. Energy, 20(11):915–920, 1966. [26] M.M.R. Williams. A simple method for calculating the linear extrapolation distance at the surface of a black body. Annals of Nuclear Energy, 15(2):101–105, 1988. [27] Vinod Kumar and S. V. G Menon. Condensation rate on a black sphere via Fokker-Planck equation. J. Chem. Phys., 82(2):917–920, 1985. [28] M. E. Widder and U. M. Titulaer. The kinetic boundary layer around an absorbing sphere and the growth of small droplets. J. Stat. Phys., 55(5/6):1109–1127, 1989. [29] S. Waldenstrøm and K. Razi Naqvi. The steady rate of absorption (by a grey sphere) of particles obeying the Fokker-Planck equation: analysis of the results obtained by moment methods. Ark. Fys. Seminar Trondheim, (4):1–37, 1988. [30] Joel H. Hildebrand. Motions of molecules in liquids: Viscosity and diffusivity. Science, 174(4008):490–493, 1971. [31] K. Razi Naqvi, K. J. Mork, and S. Waldenstrøm. Transport equations and boundary conditions for describing motion in a homogeneous solvent with a completely absorbing boundary. Ark. Fys. Seminar Trondheim, (7):1–36, 1982. [32] K. Razi Naqvi. Milne’s problem for a non-capturing medium: accurate analytic approximations for particle density and emergent angular distribution. J. Quant. Spectrosc. Radiat. Transfer, 50(1):59–64, 1993. [33] T. W. Marshall and E. J. Watson. The analytic solutions of some boundary layer problems in the theory of Brownian motion. J. Phys. A: Math. Gen., 20(6):1345–1354, 1987. This paper is available on arxiv under CC BY 4.0 DEED license. Author: (1) K. Razi Naqvi, Department of Physics, Norwegian University of Science and Technology (NTNU), 7094 Trondheim, Norway Author: Author: (1) K. Razi Naqvi, Department of Physics, Norwegian University of Science and Technology (NTNU), 7094 Trondheim, Norway Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Preliminary material 2 Preliminary material 3 Comments elicited by the solution to Milne’s problem for B-particles 3 Comments elicited by the solution to Milne’s problem for B-particles 4 Flux to a spherical trap 4 Flux to a spherical trap 5 Molecular motion in liquids 5 Molecular motion in liquids 6 Concluding remarks and References 6 Concluding remarks and References 6 Concluding remarks The claim—motivated by the search for a better boundary condition—that the Lorentz model is a useful tool for investigating bimolecular reactions in solutions became credible only after the publication of Burschka and Titulaer’s numerical solution of the one-dimensional KKE [4]. Immediately prior to that, one critic expressed the “community opinion” by stating in a referee report (on an article co-authored by me) that the LBE “is absolutely useless in dealing with transport in liquids”, and insisted that this task is best handled by solving the KKE. Only then did the need arise for comparing the length scales of inverse and regular Brownian motion. The Trondheim group has shown that inverse Brownian motion, regular Brownian motion and the BGK-model are indistinguishable at the DE-level, provided that one uses the appropriate BC [31], namely that stated in Eq. (20). A Procrustean random walk model that allows no distribution of path lengths seems (to me) unphysical, much like the lattice model that informed Smoluchowksi’s thinking (about the boundary condition at an absorbing surface), and has misinformed generations of students as well as aficionados of chemical kinetics. Infinitely heavy B-particles, infinitely light L-particles, infinitely inflexible (about the constancy of their pathlenghts) R-particles are all fictions, but some fictions are more fruitful than others, and some are outright useless. Whether the fiction of R-particles will bear fruit (in the setting of diffusion-mediated reactions) or serve as a mere distraction remains to be seen. A Milne’s problem: calculating the density profiles A Milne’s problem: calculating the density profiles The purpose of this appendix is to enable a reader of this article to generate the data used for plotting the density profiles shown in Fig. 1. The density data for L-particles were generated with the aid of a variational calculation [32], in which ℓ was used a the unit of length. The corresponding data for B-particles were computed by improving the results obtained by the Trondheim group through a half-range treatment [10], in which the 𝑁th order approximation for the particle density 𝑛 was expressed in the form and values of 𝑥0 (1.459877Λ), 𝑥𝑖 and 𝜆𝑖 (for 𝑖 =1–8) resulting from a ninth-order approximation (𝑁 = 8) were reported, and the values of 𝑛(𝑥) close to the wall were compared with those found by Marshall and Watson (M&W) on the basis of their exact analytical treatment [33]. The improvements consists of three minor changes: the value of 𝑥0 has been replaced by 𝑥0 = 1.460354Λ (the first seven figures of the exact result), one more term has been added, and the values of 𝑥𝑖 and 𝜆𝑖 for the last three terms (𝑖 =7–9) have been optimised in a least-squares fit to the numbers in column (A) of Table 1 of M&W. The complete set of {𝑥𝑖 , 𝜆𝑖} values (of mostly-analytical-partly-empirical origin) is displayed in Table 1, the upper part of which is identical with Table II of ref. [10]. For plotting the density of B-particles in Fig. 1, the length scale was changed from Λ to ℓ. References [1] M. von Smoluchowski. Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Physik. Zeit., 17:568–571 (1st instalment), 585–593 (2nd instalment), 1916. [2] M. von Smoluchowski. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z. Phys. Chem. (Leipzig), 92:129–168, 1917. [3] M. von Smoluchowski. Grundriß der Koagulationskinetik kolloider Lösungen. Kolloid Z., 21:98–104, 1917. [4] M. A. Burschka and U. M. Titulaer. The kinetic boundary layer for the Fokker-Planck equation with absorbing boundary. J. Stat. Phys., 25:569–582, 1981. [5] K. Razi Naqvi, K. J. Mork, and S. Waldenstrøm. Comment on “Steady, one-dimensional Brownian motion with an absorbing boundary”. J. Chem. Phys., 77(7):4263–4264, 1982. [6] K. Razi Naqvi, K. J. Mork, and S. Waldenstrøm. Reduction of the Fokker-Planck equation with an absorbing or reflecting boundary to the diffusion equation and the radiation boundary condition. Phys. Rev. Lett., 49:304–307, 1982. [7] K. Razi Naqvi, S. Waldenstrøm, and K. J. Mork. Kinetics of diffusion-mediated bimolecular reactions. A new theoretical framework. J. Phys. Chem., 86:4750–4756, 1982. [8] K. Razi Naqvi, S. Waldenstrøm, and K. J. Mork. On the rate of absorption of Brownian particles by a black sphere: The connection between the Fokker–Planck equation and the diffusion equation. J. Chem. Phys., 78(5):2710–2712, 1983. [9] Enrico Fermi (author) and Emilio Segre (chief editor). Collected Papers (Note E Memorie), (Vol. 1: Italy, 1921–1938). University of Chicago Press, Chicago, IL 1962. [10] K. Razi Naqvi, K. J. Mork, and S. Waldenstrm. Steady-state, one-dimensional Fokker-Planck equation with an absorbing boundary: A half-range treatment. Phys. Rev. A, 40:3405–3407, 1989. [11] S. Waldenstrøm, K. J. Mork, and K. Razi Naqvi. Milne problem for a hard-sphere Rayleigh gas: a study based on the Fokker-Planck equation. Phys. Rev. A, 28(3):1659–1661, 1983. [12] S. V. G. Menon, Vinod Kumar, and D. C. Sahni. Green’s function approach to the solution of the time dependent Fokker-Planck equation with an absorbing boundary. Physica A: Statistical Mechanics and its Applications, 135(1):63–79, 1986. [13] M. J. Lindenfeld and B. Shizgal. The Milne problem: A study of the mass dependence. Phys. Rev. A, 27:1657–1670, 1983. [14] Knud Andersen and Kurt E. Shuler. On the relaxation of the hard-sphere Rayleigh and Lorentz Gas. J. Chem. Phys., 40(3):633–650, 1964. [15] B. Davison. Neutron Transport Theory. OUP, Oxford, 1957. [16] Julian Keilson and James E. Storer. On Brownian motion, Boltzmann’s equation and the Fokker-Planck equation. Quarterly of Applied Mathematics, 10(3):243–253, 1952. [17] M. R. Hoare. The linear gas. In Ilya Prigogine and Stuart A. Rice, editors, Advance in Chemical Physics, volume 20, pages 135–214. Wiley-Interscience, New York, 1971. [18] Marie Smoluchowski. Sur le chemin moyen parcouru par les molécules d’un gaz et sur son rapport avec la théorie de la diffusion. Bull. Acad. Sci. Cracovie, pages 202–213, 1906. [19] K. Razi Naqvi. 100+ years of colossal confusion on colloidal coagulation. Part I: Smoluchowski’s work on absorbing boundaries. arXiv:2404.17021 [cond-mat.stat-mech], pages 1–20, 2024. [20] Robert M. Ziff. Flux to a trap. J. Stat. Phys., 65(5):1217–1233, 1991. [21] S. N. Majumdar, A. Comtet, and R. M. Ziff. Unified solution of the expected maximum of a discrete time random walk and the discrete flux to a spherical trap. J. Stat. Phys., 122(5):833– 856, 2006. [22] R. M. Ziff, S. N. Majumdar, and A. Comtet. General flux to a trap in one and three dimensions. J. Phys.: Condens. Matter, 19(6):065102, 2007. [23] H. L. Frisch and F. C. Collins Diffusional processes in the growth of aerosol particles. J. Chem. Phys., 20:1797–1803, 1952. [24] F. C. Collins and G. E. Kimball. Diffusion-controlled reactions in liquid solutions. Ind. Eng. Chem., 41(11):2551–2553, 1949. [25] D. C. Sahni. The effect of a black sphere on the flux distribution in an infinite moderator. J. Nucl. Energy, 20(11):915–920, 1966. [26] M.M.R. Williams. A simple method for calculating the linear extrapolation distance at the surface of a black body. Annals of Nuclear Energy, 15(2):101–105, 1988. [27] Vinod Kumar and S. V. G Menon. Condensation rate on a black sphere via Fokker-Planck equation. J. Chem. Phys., 82(2):917–920, 1985. [28] M. E. Widder and U. M. Titulaer. The kinetic boundary layer around an absorbing sphere and the growth of small droplets. J. Stat. Phys., 55(5/6):1109–1127, 1989. [29] S. Waldenstrøm and K. Razi Naqvi. The steady rate of absorption (by a grey sphere) of particles obeying the Fokker-Planck equation: analysis of the results obtained by moment methods. Ark. Fys. Seminar Trondheim, (4):1–37, 1988. [30] Joel H. Hildebrand. Motions of molecules in liquids: Viscosity and diffusivity. Science, 174(4008):490–493, 1971. [31] K. Razi Naqvi, K. J. Mork, and S. Waldenstrøm. Transport equations and boundary conditions for describing motion in a homogeneous solvent with a completely absorbing boundary. Ark. Fys. Seminar Trondheim, (7):1–36, 1982. [32] K. Razi Naqvi. Milne’s problem for a non-capturing medium: accurate analytic approximations for particle density and emergent angular distribution. J. Quant. Spectrosc. Radiat. Transfer, 50(1):59–64, 1993. [33] T. W. Marshall and E. J. Watson. The analytic solutions of some boundary layer problems in the theory of Brownian motion. J. Phys. A: Math. Gen., 20(6):1345–1354, 1987. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Revisiting Discrepancies in Diffusion Theory: Concluding Remarks and References

Author:
(1) K. Razi Naqvi, Department of Physics, Norwegian University of Science and Technology (NTNU), 7094 Trondheim, Norway Table of Links Abstract and 1 Introduction 2 Preliminary material 3 Comments elicited by the solution to Milne’s problem for B-particles 4 Flux to a spherical trap 5 Molecular motion in liquids 6 Concluding remarks and References 4 Flux to a spherical trap For absorption of R-particles by a spherical trap of radius 𝑅, Ziff found the “Milne extrapolation length” to be ≈ 0.29795219𝑙 for 0 < 𝑙 ≤ 2𝑅. Since this result contradicts (at least at first sight) a great deal of work on closely related problems, including some carried out by the present author, and no resolution was offered in Ziff’s paper [20], or in two of its sparkling sequels [21, 22], a probe into the discrepancy is warranted. We are thus led to the seemingly insipid but sublimely conciliatory conclusion that the expressions 4.1 The boundary condition at the wall 4.2 The boundary condition for a spherical sink Citing four articles by Collins and co-workers, Ziff makes two remarks, stating, first, that the “constant 𝛾 [the linear extrapolation length] must be determined by empirical arguments”, and next, that “to lowest order they [meaning various prescriptions by Collins et al] generally give the same result 𝛾 = (𝑙/3)[1 + 𝒪(𝜀)], . . . ”, where 𝜀 ≡ 𝑙/𝑅. I am unable to see an unequivocal statment to this effect in any of the four papers cited by Ziff (his refs. 15–18). In their Eqs. (3) and (4), reproduced here in my notation, Frisch and Collins [23] provide the clearest statement of their BC: Since the 𝛼-dependence is wrong (see below) and we are interested in a black sphere, we will set 𝛼 = 1 in Eq. (F&C-3). By examining ref. 15 of Ziff, a 1949 article [24], one can convince oneself that Eq. (F&C-4) should be changed to but a general statement about 𝜌 (which is to be identified with the linear extrapolation length for a black sphere) is still beyond reach because the “jumps” above discussed can be taken essentially as the path of the molecule between successive collisions. The persistence of velocity upon collision, however, causes the jump density function 𝜑(𝑠) to be no longer spherically uniform but to depend upon the direction of the jump immediately preceding the jump under consideration in the manner of a Markov process. However, . . . this effect can be accounted for by multiplying ⟨𝑠 2 ⟩ by a correction factor slightly greater than unity. For convenience in this discussion, it will be assumed that this correction factor has been already incorporated in ⟨𝑠 2 ⟩ and in the other moments of 𝜑(𝑠). For the two cases of interest to us, namely L-particles and R-particles, we get 𝜌 = 2ℓ/3 and 𝜌 = 𝑙/3, in agreement with the calculations based on the plane symmetric system. It is worth adding here that the jump model presented in the above cited 1949 article [24] can be shown to imply the following boundary condition for a grey sphere: but the inference did not appear in print until 1982 [7]. 4.3 The case of a small sphere: the Achilles heel of the “Pearson-Rayleigh random walker” So far, we have not come across any conclusion (emerging from Ziff’s model) that cannot be reconciled with its counterparts from inverse and regular Brownian motion. However, Ziff’s calculations revealed the extrapolation length 𝛾 to be independent of 𝑙/𝑅 for 0 < 𝑙/𝑅 ≤ 2, in sharp contrast to the results found in neutron transport studies, which have been summarized by Sahni [25] andWilliams [26], both of whom have presented their own calculations as well. When the Brownian analog of this problem was investigated, the coagulation rate constant showed an umistakable dependence on the value of Λ/𝑅 [27–29]; an important conclusion that emerged from these investigations is worth stressing yet again: when moment methods are used for solving the KKE, attempts to obtain better results by increasing the number of moments will prove to be futile, because, beyond a certain order, convergence is lost. This paper is available on arxiv under CC BY 4.0 DEED license. Author: (1) K. Razi Naqvi, Department of Physics, Norwegian University of Science and Technology (NTNU), 7094 Trondheim, Norway Author: Author: (1) K. Razi Naqvi, Department of Physics, Norwegian University of Science and Technology (NTNU), 7094 Trondheim, Norway Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Preliminary material 2 Preliminary material 3 Comments elicited by the solution to Milne’s problem for B-particles 3 Comments elicited by the solution to Milne’s problem for B-particles 4 Flux to a spherical trap 4 Flux to a spherical trap 5 Molecular motion in liquids 5 Molecular motion in liquids 6 Concluding remarks and References 6 Concluding remarks and References 4 Flux to a spherical trap For absorption of R-particles by a spherical trap of radius 𝑅, Ziff found the “Milne extrapolation length” to be ≈ 0.29795219𝑙 for 0 < 𝑙 ≤ 2𝑅. Since this result contradicts (at least at first sight) a great deal of work on closely related problems, including some carried out by the present author, and no resolution was offered in Ziff’s paper [20], or in two of its sparkling sequels [21, 22], a probe into the discrepancy is warranted. We are thus led to the seemingly insipid but sublimely conciliatory conclusion that the expressions 4.1 The boundary condition at the wall 4.2 The boundary condition for a spherical sink Citing four articles by Collins and co-workers, Ziff makes two remarks, stating, first, that the “constant 𝛾 [the linear extrapolation length] must be determined by empirical arguments”, and next, that “to lowest order they [meaning various prescriptions by Collins et al] generally give the same result 𝛾 = (𝑙/3)[1 + 𝒪(𝜀)], . . . ”, where 𝜀 ≡ 𝑙/𝑅. I am unable to see an unequivocal statment to this effect in any of the four papers cited by Ziff (his refs. 15–18). In their Eqs. (3) and (4), reproduced here in my notation, Frisch and Collins [23] provide the clearest statement of their BC: Since the 𝛼-dependence is wrong (see below) and we are interested in a black sphere, we will set 𝛼 = 1 in Eq. (F&C-3). By examining ref. 15 of Ziff, a 1949 article [24], one can convince oneself that Eq. (F&C-4) should be changed to but a general statement about 𝜌 (which is to be identified with the linear extrapolation length for a black sphere) is still beyond reach because the “jumps” above discussed can be taken essentially as the path of the molecule between successive collisions. The persistence of velocity upon collision, however, causes the jump density function 𝜑(𝑠) to be no longer spherically uniform but to depend upon the direction of the jump immediately preceding the jump under consideration in the manner of a Markov process. However, . . . this effect can be accounted for by multiplying ⟨𝑠 2 ⟩ by a correction factor slightly greater than unity. For convenience in this discussion, it will be assumed that this correction factor has been already incorporated in ⟨𝑠 2 ⟩ and in the other moments of 𝜑(𝑠). For the two cases of interest to us, namely L-particles and R-particles, we get 𝜌 = 2ℓ/3 and 𝜌 = 𝑙/3, in agreement with the calculations based on the plane symmetric system. It is worth adding here that the jump model presented in the above cited 1949 article [24] can be shown to imply the following boundary condition for a grey sphere: but the inference did not appear in print until 1982 [7]. 4.3 The case of a small sphere: the Achilles heel of the “Pearson-Rayleigh random walker” So far, we have not come across any conclusion (emerging from Ziff’s model) that cannot be reconciled with its counterparts from inverse and regular Brownian motion. However, Ziff’s calculations revealed the extrapolation length 𝛾 to be independent of 𝑙/𝑅 for 0 < 𝑙/𝑅 ≤ 2, in sharp contrast to the results found in neutron transport studies, which have been summarized by Sahni [25] andWilliams [26], both of whom have presented their own calculations as well. When the Brownian analog of this problem was investigated, the coagulation rate constant showed an umistakable dependence on the value of Λ/𝑅 [27–29]; an important conclusion that emerged from these investigations is worth stressing yet again: when moment methods are used for solving the KKE, attempts to obtain better results by increasing the number of moments will prove to be futile, because, beyond a certain order, convergence is lost. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Revisiting Discrepancies in Diffusion Theory: Flux to a Spherical Trap

Author:
(1) K. Razi Naqvi, Department of Physics, Norwegian University of Science and Technology (NTNU), 7094 Trondheim, Norway Table of Links Abstract and 1 Introduction 2 Preliminary material 3 Comments elicited by the solution to Milne’s problem for B-particles 4 Flux to a spherical trap 5 Molecular motion in liquids 6 Concluding remarks and References In a paper on “the Brownian motion analog of the well-known Milne problem in radiative transfer theory” [J Stat Phys 25 (1981) 569–82], Burschka and Titulaer reported: “The value we find for this ‘Milne extrapolation length’ is, in the appropriate dimensionless units, approximately twice the value found in the radiative transfer problem.” A study by Ziff [J Stat Phys 65 (1991) 1217–33], concerned with the absorption of particles executing a Rayleigh flight (randomly directed displacements of equal length 𝑙) by a black sphere of radius 𝑅, led to a value for the extrapolation length 𝛾 about half as small as the benchmark result (for 𝑅 ≫ 𝑙). The first discrepancy is shown to result from the disparity of the two length scales; the second, from the zero variance of the jump lengths. Ziff’s finding that 𝛾 is independent of 𝑅 for 0 < 𝑙 ≤ 2𝑅, cannot be reconciled with studies based on the Lorentz-Boltzmann equation and the Klein-Kramers equation. 1 Introduction The purpose of this article is to ask which if either of the discrepancies mentioned in the abstract is genuine, and to probe into their implications for the long-established (but illadvised) practice of equating the rate constant of diffusion-mediated bimolecular reactions in liquids to the flux found by solving the diffusion equation (DE) subject to the Smoluchowski boundary condition (SBC), and multiplying the calculated flux–if found to be smaller than the experimentally determined rate constant–by a factor smaller than unity [1–3]. The background can be best laid out by quoting the introductory lines from a 1981 article [4] of Burschka and Titulaer (B&T): The flow of a reactant in a diffusion-controlled reaction can often be described in terms of Brownian motion of a particle in the presence of absorbing or partially absorbing boundaries. The simplest description is obtained through a diffusion or Smoluchowski equation for the probability density of the particle position with either absorbing or “radiative” boundary conditions. In the former case the density is put equal to zero at the boundary, while in the latter the outward normal flux is proportional to the density with a phenomenological proportionality constant. This traditional theory has often been criticized; in particular there seems to be no clear way of relating the proportionality constant in the radiative boundary conditions to a microscopic picture of the reaction kinetics. The reasons for the inadequacy of the Smoluchowski equation can be exhibited by inspecting its derivation from a more detailed description of Brownian motion due to Klein and Kramers, in terms of the probability density for the velocity and position of the Brownian particle. The Smoluchowski equation can be recovered from this description via a procedure of the Chapman-Enskog type. This derivation breaks down, however, near a wall or at places where the potential varies rapidly; there a so-called kinetic boundary layer occurs. This breakdown is caused by the large deviations from the Maxwellian velocity distribution that must occur, e.g., near an absorbing boundary, whereas for validity of the Smoluchowski equation approximate local equilibrium is required. [Bibliographic indicators have been suppressed here.] The two paragraphs quoted above provide an accurate account of the (then) prevalent paradigm. B&T could not have foreseen that new evidence, sufficient to overturn the paradigm, was in the offing. The evidence [5– 8], which became available soon after the publication of ref [4], failed to deflate the paradigm, and caused at most a few minute punctures, which appeared to have been repaired by the dust which accumulates on printed matter and memory cells, if not blown away every now and again by enquiring spirits. Some comments on the passage are therefore in order, and should be construed not as shooting the messenger but as an attempt to shoot down a paradigm that has proved to be particularly recalcitrant. Although an additional and shorter excerpt from ref [4] is needed, a statement of the Milne problem (tailored for this note), a few nomenclatory notes, and a recapitulation of the results most pertinent for the present discussion must precede the last excerpt. This paper is available on arxiv under CC BY 4.0 DEED license. Author: (1) K. Razi Naqvi, Department of Physics, Norwegian University of Science and Technology (NTNU), 7094 Trondheim, Norway Author: Author: (1) K. Razi Naqvi, Department of Physics, Norwegian University of Science and Technology (NTNU), 7094 Trondheim, Norway Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Preliminary material 2 Preliminary material 3 Comments elicited by the solution to Milne’s problem for B-particles 3 Comments elicited by the solution to Milne’s problem for B-particles 4 Flux to a spherical trap 4 Flux to a spherical trap 5 Molecular motion in liquids 5 Molecular motion in liquids 6 Concluding remarks and References 6 Concluding remarks and References In a paper on “the Brownian motion analog of the well-known Milne problem in radiative transfer theory” [J Stat Phys 25 (1981) 569–82], Burschka and Titulaer reported: “The value we find for this ‘Milne extrapolation length’ is, in the appropriate dimensionless units, approximately twice the value found in the radiative transfer problem.” A study by Ziff [J Stat Phys 65 (1991) 1217–33], concerned with the absorption of particles executing a Rayleigh flight (randomly directed displacements of equal length 𝑙) by a black sphere of radius 𝑅, led to a value for the extrapolation length 𝛾 about half as small as the benchmark result (for 𝑅 ≫ 𝑙). The first discrepancy is shown to result from the disparity of the two length scales; the second, from the zero variance of the jump lengths. Ziff’s finding that 𝛾 is independent of 𝑅 for 0 < 𝑙 ≤ 2𝑅, cannot be reconciled with studies based on the Lorentz-Boltzmann equation and the Klein-Kramers equation. 1 Introduction The purpose of this article is to ask which if either of the discrepancies mentioned in the abstract is genuine, and to probe into their implications for the long-established (but illadvised) practice of equating the rate constant of diffusion-mediated bimolecular reactions in liquids to the flux found by solving the diffusion equation (DE) subject to the Smoluchowski boundary condition (SBC), and multiplying the calculated flux–if found to be smaller than the experimentally determined rate constant–by a factor smaller than unity [1–3]. The background can be best laid out by quoting the introductory lines from a 1981 article [4] of Burschka and Titulaer (B&T): The flow of a reactant in a diffusion-controlled reaction can often be described in terms of Brownian motion of a particle in the presence of absorbing or partially absorbing boundaries. The simplest description is obtained through a diffusion or Smoluchowski equation for the probability density of the particle position with either absorbing or “radiative” boundary conditions. In the former case the density is put equal to zero at the boundary, while in the latter the outward normal flux is proportional to the density with a phenomenological proportionality constant. This traditional theory has often been criticized; in particular there seems to be no clear way of relating the proportionality constant in the radiative boundary conditions to a microscopic picture of the reaction kinetics. The reasons for the inadequacy of the Smoluchowski equation can be exhibited by inspecting its derivation from a more detailed description of Brownian motion due to Klein and Kramers, in terms of the probability density for the velocity and position of the Brownian particle. The Smoluchowski equation can be recovered from this description via a procedure of the Chapman-Enskog type. This derivation breaks down, however, near a wall or at places where the potential varies rapidly; there a so-called kinetic boundary layer occurs. This breakdown is caused by the large deviations from the Maxwellian velocity distribution that must occur, e.g., near an absorbing boundary, whereas for validity of the Smoluchowski equation approximate local equilibrium is required. [Bibliographic indicators have been suppressed here.] The two paragraphs quoted above provide an accurate account of the (then) prevalent paradigm. B&T could not have foreseen that new evidence, sufficient to overturn the paradigm, was in the offing. The evidence [5– 8], which became available soon after the publication of ref [4], failed to deflate the paradigm, and caused at most a few minute punctures, which appeared to have been repaired by the dust which accumulates on printed matter and memory cells, if not blown away every now and again by enquiring spirits. Some comments on the passage are therefore in order, and should be construed not as shooting the messenger but as an attempt to shoot down a paradigm that has proved to be particularly recalcitrant. Although an additional and shorter excerpt from ref [4] is needed, a statement of the Milne problem (tailored for this note), a few nomenclatory notes, and a recapitulation of the results most pertinent for the present discussion must precede the last excerpt. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Revisiting Discrepancies in Diffusion Theory: Abstract and Introduction

Machine Learning Enhanced Characterization of Labyrinthine Structures in Bi-doped Films

This paper is available on arxiv under CC 4.0 license. Authors: (1) Eleonora Alei, ETH Zurich, Institute for Particle Physics & Astrophysics & National Center of Competence in Research PlanetS; (2) Björn S. Konrad, ETH Zurich, Institute for Particle Physics & Astrophysics & National Center of Competence in Research PlanetS; (3) Daniel Angerhausen, ETH Zurich, Institute for Particle Physics & Astrophysics, National Center of Competence in Research PlanetS & Blue Marble Space Institute of Science; (4) John Lee Grenfell, Department of Extrasolar Planets and Atmospheres (EPA), Institute for Planetary Research (PF), German Aerospace Centre (DLR) (5) Paul Mollière, Max-Planck-Institut für Astronomie; (6) Sascha P. Quanz, ETH Zurich, Institute for Particle Physics & Astrophysics & National Center of Competence in Research PlanetS; (7) Sarah Rugheimer, Department of Physics, University of Oxford; (8) Fabian Wunderlich, Department of Extrasolar Planets and Atmospheres (EPA), Institute for Planetary Research (PF), German Aerospace Centre (DLR); (9) LIFE collaboration, www.life-space-mission.com. Table of Links Abstract & Introduction Methods Results Discussion Conclusions Next Steps & References Appendix A: Scattering of terrestrial exoplanets Appendix B: Corner Plots Appendix C: Bayes’ factor analysis: other epochs Appendix D: Cloudy scenarios: additional figures Appendix C: Bayes’ factor analysis: other epochs As described in Section 2, the theoretical spectral model was updated with respect to Paper III and it now takes into account additional physical processes. For the results presented in Section 3, we ran retrievals using the most updated version of the Bayesian framework. The additional flexibility of petitRADTRANS now allows us to quantify the impact of CIA and scattering in retrievals. We ran additional retrievals on the clear sky scenarios for R = 50 and S/N = 10. In these retrievals, we altered the number of physical processes that were treated in the petitRADTRANS theoretical spectral model as follows: – Including both CIA and scattering (setup used in Section 3); – Excluding both CIA and scattering; – Including scattering and excluding CIA; – Including CIA and excluding scattering. In the runs where scattering is included, we consider selfscattering, surface scattering of the thermal radiation, and gaseous Rayleigh scattering (see Table 5 for references). We do not include aerosol and cloud scattering in the calculation. Since our theoretical spectral model neglects clouds, in this analysis we considered only the cloud-free scenarios. The effect of modeling cloudy spectra using a cloud-free retrieval model will be discussed in detail in the Section 4.2. To determine the theoretical spectral model configuration that best reproduces the input spectra we performed a Bayes’ factor analysis. The Bayes’ factor is defined as: The results for the remaining epochs exhibit similar behaviour. In Figure C.2, we show the results for the NOE Earth, Figure C.3 shows the ones for the GOE Earth, and the results for the prebiotic Earth are shown in Figure C.4. This paper is available on under CC 4.0 license. arxiv

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