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CHICAGO (AP) — Cam Fowler scored twice in his 1,000th NHL game, and the St. Louis Blues beat the Chicago Blackhawks 6-2 on Tuesday in the Winter Classic at Wrigley Field.

Blues beat the Blackhawks in the Winter Classic - Yahoo Sports

Location: Lincoln Financial Field, Philadelphia, PA

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CLEVELAND, June 24, 2022 /PRNewswire/ -- Downtown Cleveland gains a new Rally House store in the historic Caxton Building within the Gateway District. Rally House Gateway is conveniently situated down the street from Rocket Mortgage FieldHouse, Progressive Field, and FirstEnergy Stadium, a ...

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The kick is the longest made field goal at the top level of college football since 2008 when UTEP’s Jose Martinez hit a 64-yard field goal 16 years ago. It’s also the longest kick in the history of Lincoln Financial Field in either a college football or NFL game.

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The business, CPJ Field, employs 147 people and is typical of the kind of British family enterprise that gets handed down through the decades.

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Cousins mounted a quick 70-yard touchdown drive that lasted just 65 seconds to lift the Atlanta Falcons to a 22-21 win over the Eagles at Lincoln Financial Field.

Kirk Cousins, Falcons mount late game-winning drive to stun Eagles, grab 22-21 win on 'Monday Night Football' - Yahoo Sports

Field

field

Is the Universe a Self-Knowing System? The Case for the Wisdom Field For centuries, we’ve assumed that reality is made of matter and energy—that space-time is the fundamental backdrop of existence. But what if we’ve had it backwards? What if space-time, matter, and even consciousness emerge from something deeper—a fundamental informational field, a Wisdom Field? Quantum mechanics, neuroscience, and complexity science all point in the same direction: information is the foundation of reality. And intelligence, selfhood, and meaning? They’re just progressive refinements of that same informational structure. This article breaks down why this theory makes sense, what evidence supports it, and why it changes everything. The Universe is Information, Not Stuff Physics has long struggled with the weirdness of quantum mechanics. Entangled particles seem to affect each other instantly, violating our sense of space-time. Observing a quantum system collapses its wave function as if reality depends on measurement. But this isn’t strange if we consider a radical idea: Space-time is not fundamental. Information is. Quantum entanglement isn’t “spooky action at a distance” if distance itself is an emergent product of information structure. Selfhood: An Informational Loop Neuroscience is also revealing something profound: consciousness isn’t just a byproduct of neurons firing. It’s an emergent property of information processing. The brain is not just a storage device—it is an active organizer of information, structuring data into meaning. Complexity research shows that consciousness scales with the amount of integrated information. The brain doesn’t generate selfhood—it localizes and refines it. In other words, selfhood is what happens when information loops in on itself in a structured way. Knowledge as Nested Analogies But what about knowledge? How do we “know” anything? It turns out, all knowledge may just be nested analogies. Every idea, every concept, every theory is a structured web of comparisons that grows in complexity over time. Consider how science progresses: Newton’s gravity → Einstein’s relativity → Quantum mechanics → ??? Each step isn’t a rejection of the previous—it’s a deeper refinement of analogy. Over laps on the previous and is understood from its use of historical symbols. Layer upon layer. Knowing emerges as patterns of analogy reinforced and stabilized. The deeper the analogy structure, the greater the knowing. The Wisdom Field: A Self-Knowing Universe If all of this is true, then we’re left with one staggering conclusion: ➡️ The universe isn’t just a collection of things—it is a self-organizing, self-knowing system. ➡️ Wisdom isn’t a property of the universe. Wisdom is the universe. Everything—space-time, matter, intelligence—is just different expressions of a fundamental informational field organizing itself into complexity. And eventually knowledge. All the same stuff but so different. Is the Universe a Self-Knowing System? The Case for the Wisdom Field For centuries, we’ve assumed that reality is made of matter and energy—that space-time is the fundamental backdrop of existence. But what if we’ve had it backwards? What if space-time, matter, and even consciousness emerge from something deeper—a fundamental informational field, a Wisdom Field? Quantum mechanics, neuroscience, and complexity science all point in the same direction: information is the foundation of reality. And intelligence, selfhood, and meaning? They’re just progressive refinements of that same informational structure. This article breaks down why this theory makes sense, what evidence supports it, and why it changes everything. The Universe is Information, Not Stuff The Universe is Information, Not Stuff Physics has long struggled with the weirdness of quantum mechanics. Entangled particles seem to affect each other instantly, violating our sense of space-time. Observing a quantum system collapses its wave function as if reality depends on measurement. But this isn’t strange if we consider a radical idea: Space-time is not fundamental. Information is. Quantum entanglement isn’t “spooky action at a distance” if distance itself is an emergent product of information structure. Selfhood: An Informational Loop Selfhood: An Informational Loop Neuroscience is also revealing something profound: consciousness isn’t just a byproduct of neurons firing. It’s an emergent property of information processing. The brain is not just a storage device—it is an active organizer of information, structuring data into meaning. Complexity research shows that consciousness scales with the amount of integrated information. The brain doesn’t generate selfhood—it localizes and refines it. In other words, selfhood is what happens when information loops in on itself in a structured way. Knowledge as Nested Analogies Knowledge as Nested Analogies But what about knowledge? How do we “know” anything? It turns out, all knowledge may just be nested analogies. Every idea, every concept, every theory is a structured web of comparisons that grows in complexity over time. Consider how science progresses: Newton’s gravity → Einstein’s relativity → Quantum mechanics → ??? Each step isn’t a rejection of the previous—it’s a deeper refinement of analogy. Over laps on the previous and is understood from its use of historical symbols. Layer upon layer. Knowing emerges as patterns of analogy reinforced and stabilized. The deeper the analogy structure, the greater the knowing. The Wisdom Field: A Self-Knowing Universe The Wisdom Field: A Self-Knowing Universe If all of this is true, then we’re left with one staggering conclusion: ➡️ The universe isn’t just a collection of things—it is a self-organizing, self-knowing system. ➡️ Wisdom isn’t a property of the universe. Wisdom is the universe. Everything—space-time, matter, intelligence—is just different expressions of a fundamental informational field organizing itself into complexity. And eventually knowledge. All the same stuff but so different.

Wisdom Field - A Theory of Everything 

Authors:
(1) Nandan Pakhira, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India;
(2) Rajib Mahato, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India and Central Electronics Engineering Research Institute, Pilani, Rajasthan 333031, India. Table of Links Abstract and 1 Introduction II. Mathematical Formalism III. Work Function Distribution IV. Results A. Gaussian work function distribution B. Log-normal work function distribution V. Conclusion, Acknowledgements, and References I. INTRODUCTION Field emission is the process in which electrons from cold surfaces are emitted in the presence of applied strong electric field. This process should be compared agaist the thermo-ionic process in which electrons are emitted from hot metal surface. The field emission forms the back bone of modern semiconductor devices. This effect was first described by Fowler and Nordheim[1]. They considered quantum mechanical tunneling through a triangular potential energy (PE) barrier, created by the application of a constant electric field. Much later Murphy and Good[2] (MG) introduced a more realistic PE barrier by taking into account the induced image charge formed in the presence of emitted electrons. MG calculated the barrier transmission coefficient under semi-classical WKB approximation. More recently, essentially an exact solution of the problem was obtained by Choy et. al.[3]. Also, various cases including finite temperature (thermal emission), tunneling, curvature of th. To the best of our knowledge in all of those studies authors have considered constant local work function. The work function of a material depends on the composition, structure, geometry, local charge distribution etc. of the emitting surface. Assumption of constant work function is only suitable for an atomistic smooth homogeneous surface. For surfaces with inhomogeneities over nano-scale (much smaller than the size of the collectors) assumption of constant work function is no longer valid. Also it has been shown[12] that in a system of nanoparticles there is a distribution of the size of the nanoparticles. Since the work function is more of a property of the surface we naturally can expect that the work function of nano-particles will also have a distribution. The actual microscopic model for work function distribution for a system of nano-particles is beyond the scope of this work. Interestingly, Gamez et. al.[12] using scanning tunneling microscope (STM) have measured the pair distribution function (PDF) for Pt nano-particles and they found that it follows log-normal distribution. In this work, purely as a mathematical model, we choose Gaussian and log-normal distribution for the work function. We then study the field emision current averaged over work function distribution. The organization of the rest of the paper is as follows. In Sec. II we describe the mathematical formalism used to calculate field emission current. In Sec. III we describe the work function distribution used to calculate average current. In Sec. IV we present our results for both the case of Gaussian distribution and log-normal distribution. Finally in Sec. V we conclude. II. MATHEMATICAL FORMALISM We closely follow and summarize the results obtained by Lopes et. al.[13],[14] for field-emission current density, J. In the standard FN-type MG theory the field emission current density is given by the well-known expression[15–18], K(λ) and E(λ) are the complete elliptic integral of the first and second kind, with From the relations above it is quite evident that calculation of transmission current requires evaluation of numerical integrals for complete elliptic integrals. Due to the singularities present in complete elliptic integrals, it is very hard to extract meaningful results purely numerically[19]. Under this circumstances we can consider series expansion for complete elliptic integrals[20] as follows where ls A detailed calculation gives the following form for the field emission current density This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Nandan Pakhira, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India; (2) Rajib Mahato, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India and Central Electronics Engineering Research Institute, Pilani, Rajasthan 333031, India. Authors: Authors: (1) Nandan Pakhira, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India; (2) Rajib Mahato, Department of Physics, Kazi Nazrul University, Asansol, West Bengal 713340, India and Central Electronics Engineering Research Institute, Pilani, Rajasthan 333031, India. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction II. Mathematical Formalism II. Mathematical Formalism III. Work Function Distribution III. Work Function Distribution IV. Results A. Gaussian work function distribution A. Gaussian work function distribution B. Log-normal work function distribution B. Log-normal work function distribution V. Conclusion, Acknowledgements, and References V. Conclusion, Acknowledgements, and References I. INTRODUCTION Field emission is the process in which electrons from cold surfaces are emitted in the presence of applied strong electric field. This process should be compared agaist the thermo-ionic process in which electrons are emitted from hot metal surface. The field emission forms the back bone of modern semiconductor devices. This effect was first described by Fowler and Nordheim[1]. They considered quantum mechanical tunneling through a triangular potential energy (PE) barrier, created by the application of a constant electric field. Much later Murphy and Good[2] (MG) introduced a more realistic PE barrier by taking into account the induced image charge formed in the presence of emitted electrons. MG calculated the barrier transmission coefficient under semi-classical WKB approximation. More recently, essentially an exact solution of the problem was obtained by Choy et. al.[3]. Also, various cases including finite temperature (thermal emission), tunneling, curvature of th. To the best of our knowledge in all of those studies authors have considered constant local work function. The work function of a material depends on the composition, structure, geometry, local charge distribution etc. of the emitting surface. Assumption of constant work function is only suitable for an atomistic smooth homogeneous surface. For surfaces with inhomogeneities over nano-scale (much smaller than the size of the collectors) assumption of constant work function is no longer valid. Also it has been shown[12] that in a system of nanoparticles there is a distribution of the size of the nanoparticles. Since the work function is more of a property of the surface we naturally can expect that the work function of nano-particles will also have a distribution. The actual microscopic model for work function distribution for a system of nano-particles is beyond the scope of this work. Interestingly, Gamez et. al.[12] using scanning tunneling microscope (STM) have measured the pair distribution function (PDF) for Pt nano-particles and they found that it follows log-normal distribution. In this work, purely as a mathematical model, we choose Gaussian and log-normal distribution for the work function. We then study the field emision current averaged over work function distribution. The organization of the rest of the paper is as follows. In Sec. II we describe the mathematical formalism used to calculate field emission current. In Sec. III we describe the work function distribution used to calculate average current. In Sec. IV we present our results for both the case of Gaussian distribution and log-normal distribution. Finally in Sec. V we conclude. II. MATHEMATICAL FORMALISM We closely follow and summarize the results obtained by Lopes et. al.[13],[14] for field-emission current density, J. In the standard FN-type MG theory the field emission current density is given by the well-known expression[15–18], K(λ) and E(λ) are the complete elliptic integral of the first and second kind, with From the relations above it is quite evident that calculation of transmission current requires evaluation of numerical integrals for complete elliptic integrals. Due to the singularities present in complete elliptic integrals, it is very hard to extract meaningful results purely numerically[19]. Under this circumstances we can consider series expansion for complete elliptic integrals[20] as follows where ls A detailed calculation gives the following form for the field emission current density 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

Unlocking the Mystery of Nano-Powered Field Emission

How DeepSee Is Changing the Way Scientists Use Data in the Field

Table of Links Abstract and 1 Introduction 2 Related Work 3 Methodology 4 Studying Deep Ocean Ecosystem and 4.1 Deep Ocean Research Goals 4.2 Workflow and Data 4.3 Design Challenges and User Tasks 5 The DeepSea System 5.1 Map View
5.2 Core View 5.3 Interpolation View and 5.4 Implementation 6 Usage Scenarios and 6.1 Scenario: Pre-Cruise Planning 6.2 Scenario: On-the-Fly Decision-Making 7 Evaluation and 7.1 Cruise Deployment 7.2 Expert Interviews 7.3 Limitations 7.4 Lessons Learned 8 Conclusions and Future Work, Acknowledgments, and References 3 METHODOLOGY In this paper, we followed Sedlmair et al.’s methodological framework for design studies [43]. Each phase of the study resulted in a contribution, namely: (1) a characterization of the problem domain via design challenges and user tasks; (2) a novel and validated visualization design; and (3) reflections on the process and lessons learned for improving visualization and interaction design guidelines. We connect these phases throughout the paper by illustrating how our synthesized challenges and tasks informed DeepSee’s design and how DeepSee’s success in addressing the challenges and tasks is reflected in our evaluation. To the best of our knowledge, the problem we describe in Sect. 4 is unique to our collaborators and has not been addressed by other researchers so far. Thus, we posit that a design study based on active participation from collaborators in both the design and evaluation phases is necessary to successfully support domain experts in their highly specialized tasks. We began by establishing a close collaboration between visualization researchers and domain experts, to properly assess the impact of our approach on solving a real-world problem. In line with prior design studies [15, 18, 33], our team comprised all authors and included visualization experts in human-centered computing, interaction design, scientific data visualization and art, as well as scientists and researchers with expertise in environmental microbiology, geochemistry and geology. We developed DeepSee through several user-centered design methodologies including contextual inquiry, mixed-fidelity prototyping, user testing, and design iteration. Over an intensive ten-week period, we worked closely together to develop a shared understanding of the importance and aims of their scientific research (Sect. 4.1), characterize data and workflows (Sect. 4.2), and identify data visualization challenges and user tasks (Sect. 4.3) that would guide our system design. Our process started with contextual inquiry, combining observation and interviews with our domain collaborators to deeply understand their existing tools and workflows, discuss unmet needs and pain points, and identify key technical constraints. After two weeks, we began mixed-fidelity prototyping, starting with paper prototypes, wireframes and small interactive prototypes in code, before moving on to progressively higher-fidelity prototypes. These prototypes were iteratively discussed with the science team over the next seven weeks through semi-structured user testing. Using these prototypes as stimuli, we deepened our understanding of the problem space through on-going contextual inquiry. During this time, systems diagrams and conceptual workflows were also created to help us validate our understanding of the problem space in collaboration with the science team. As a result of this approach, we progressively uncovered insights that guided each subsequent design iteration, leading us to generate a working prototype described in Sect. 5 that addressed our collaborators’ tasks. DeepSee was then deployed in the wild [43] to assess whether our approach solves a real problem – bridging gaps in supporting visual analysis of samples during field research. Deploying DeepSee with the same experts that we designed with allowed us to assess whether domain experts are indeed helped by the new solution, as well as reflect on and refine principles for visualization and interaction design guidelines [43]. Specifically, we investigated the impact of introducing data visualizations into both planning before fieldwork as well as tactical decision making in the field. We learned that visualizing data at multiple scales, integrating 2D and 3D data in the same interface, and modularizing visualizations positively impacted scientists planning workflows the most, and that additional support for integrating data on the fly could better enable in-situ decision making. Methodological details of our evaluation and how we synthesized these lessons learned are provided in Sect. 7. This paper is available on arxiv under CC BY 4.0 DEED license. Authors:
(1) Adam Coscia, Georgia Institute of Technology, Atlanta, Georgia, USA (acoscia6@gatech.edu);
(2) Haley M. Sapers, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USA (hsapers@caltech.edu);
(3) Noah Deutsch, Harvard University Cambridge, Massachusetts, USA (ndeutsch@mde.harvard.edu);
(4) Malika Khurana, The New York Times Company, New York, New York, USA (malika.khurana@nytimes.com);
(5) John S. Magyar, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (jmagyar@caltech.edu);
(6) Sergio A. Parra, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (sparra@caltech.edu);
(7) Daniel R. Utter, rwipfler@caltech.edu Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (dutter@caltech.edu);
(8) John S. Magyar, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (rwipfler@caltech.edu);
(9) David W. Caress, Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (caress@mbari.org);
(10) Eric J. Martin Jennifer B. Paduan Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (emartin@mbari.org);
(11) Jennifer B. Paduan, Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (paje@mbari.org);
(12) Maggie Hendrie, ArtCenter College of Design, Pasadena, California, USA (maggie.hendrie@artcenter.edu);
(13) Santiago Lombeyda, California Institute of Technology, Pasadena, California, USA (santiago@caltech.edu);
(14) Hillary Mushkin, California Institute of Technology, Pasadena, California, USA (hmushkin@caltech.edu);
(15) Alex Endert, Georgia Institute of Technology, Atlanta, Georgia, USA (endert@gatech.edu);
(16) Scott Davidoff, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA (scott.davidoff@jpl.nasa.gov);
(17) Victoria J. Orphan, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USA (vorphan@caltech.edu). Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Related Work 2 Related Work 3 Methodology 3 Methodology 4 Studying Deep Ocean Ecosystem and 4.1 Deep Ocean Research Goals 4 Studying Deep Ocean Ecosystem and 4.1 Deep Ocean Research Goals 4.2 Workflow and Data 4.2 Workflow and Data 4.3 Design Challenges and User Tasks 4.3 Design Challenges and User Tasks 5 The DeepSea System 5 The DeepSea System 5.1 Map View 5.2 Core View 5.1 Map View 5.2 Core View 5.3 Interpolation View and 5.4 Implementation 5.3 Interpolation View and 5.4 Implementation 6 Usage Scenarios and 6.1 Scenario: Pre-Cruise Planning 6 Usage Scenarios and 6.1 Scenario: Pre-Cruise Planning 6.2 Scenario: On-the-Fly Decision-Making 6.2 Scenario: On-the-Fly Decision-Making 7 Evaluation and 7.1 Cruise Deployment 7 Evaluation and 7.1 Cruise Deployment 7.2 Expert Interviews 7.2 Expert Interviews 7.3 Limitations 7.3 Limitations 7.4 Lessons Learned 7.4 Lessons Learned 8 Conclusions and Future Work, Acknowledgments, and References 8 Conclusions and Future Work, Acknowledgments, and References 3 METHODOLOGY In this paper, we followed Sedlmair et al.’s methodological framework for design studies [43]. Each phase of the study resulted in a contribution, namely: (1) a characterization of the problem domain via design challenges and user tasks; (2) a novel and validated visualization design; and (3) reflections on the process and lessons learned for improving visualization and interaction design guidelines. We connect these phases throughout the paper by illustrating how our synthesized challenges and tasks informed DeepSee’s design and how DeepSee’s success in addressing the challenges and tasks is reflected in our evaluation. To the best of our knowledge, the problem we describe in Sect. 4 is unique to our collaborators and has not been addressed by other researchers so far. Thus, we posit that a design study based on active participation from collaborators in both the design and evaluation phases is necessary to successfully support domain experts in their highly specialized tasks. We began by establishing a close collaboration between visualization researchers and domain experts, to properly assess the impact of our approach on solving a real-world problem. In line with prior design studies [15, 18, 33], our team comprised all authors and included visualization experts in human-centered computing, interaction design, scientific data visualization and art, as well as scientists and researchers with expertise in environmental microbiology, geochemistry and geology. We developed DeepSee through several user-centered design methodologies including contextual inquiry, mixed-fidelity prototyping, user testing, and design iteration. Over an intensive ten-week period, we worked closely together to develop a shared understanding of the importance and aims of their scientific research (Sect. 4.1), characterize data and workflows (Sect. 4.2), and identify data visualization challenges and user tasks (Sect. 4.3) that would guide our system design. Our process started with contextual inquiry, combining observation and interviews with our domain collaborators to deeply understand their existing tools and workflows, discuss unmet needs and pain points, and identify key technical constraints. After two weeks, we began mixed-fidelity prototyping, starting with paper prototypes, wireframes and small interactive prototypes in code, before moving on to progressively higher-fidelity prototypes. These prototypes were iteratively discussed with the science team over the next seven weeks through semi-structured user testing. Using these prototypes as stimuli, we deepened our understanding of the problem space through on-going contextual inquiry. During this time, systems diagrams and conceptual workflows were also created to help us validate our understanding of the problem space in collaboration with the science team. As a result of this approach, we progressively uncovered insights that guided each subsequent design iteration, leading us to generate a working prototype described in Sect. 5 that addressed our collaborators’ tasks. DeepSee was then deployed in the wild [43] to assess whether our approach solves a real problem – bridging gaps in supporting visual analysis of samples during field research. Deploying DeepSee with the same experts that we designed with allowed us to assess whether domain experts are indeed helped by the new solution, as well as reflect on and refine principles for visualization and interaction design guidelines [43]. Specifically, we investigated the impact of introducing data visualizations into both planning before fieldwork as well as tactical decision making in the field. We learned that visualizing data at multiple scales, integrating 2D and 3D data in the same interface, and modularizing visualizations positively impacted scientists planning workflows the most, and that additional support for integrating data on the fly could better enable in-situ decision making. Methodological details of our evaluation and how we synthesized these lessons learned are provided in Sect. 7. 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 Authors: (1) Adam Coscia, Georgia Institute of Technology, Atlanta, Georgia, USA (acoscia6@gatech.edu); (2) Haley M. Sapers, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USA (hsapers@caltech.edu); (3) Noah Deutsch, Harvard University Cambridge, Massachusetts, USA (ndeutsch@mde.harvard.edu); (4) Malika Khurana, The New York Times Company, New York, New York, USA (malika.khurana@nytimes.com); (5) John S. Magyar, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (jmagyar@caltech.edu); (6) Sergio A. Parra, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (sparra@caltech.edu); (7) Daniel R. Utter, rwipfler@caltech.edu Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (dutter@caltech.edu); (8) John S. Magyar, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (rwipfler@caltech.edu); (9) David W. Caress, Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (caress@mbari.org); (10) Eric J. Martin Jennifer B. Paduan Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (emartin@mbari.org); (11) Jennifer B. Paduan, Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (paje@mbari.org); (12) Maggie Hendrie, ArtCenter College of Design, Pasadena, California, USA (maggie.hendrie@artcenter.edu); (13) Santiago Lombeyda, California Institute of Technology, Pasadena, California, USA (santiago@caltech.edu); (14) Hillary Mushkin, California Institute of Technology, Pasadena, California, USA (hmushkin@caltech.edu); (15) Alex Endert, Georgia Institute of Technology, Atlanta, Georgia, USA (endert@gatech.edu); (16) Scott Davidoff, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA (scott.davidoff@jpl.nasa.gov); (17) Victoria J. Orphan, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USA (vorphan@caltech.edu). Authors: Authors: (1) Adam Coscia, Georgia Institute of Technology, Atlanta, Georgia, USA (acoscia6@gatech.edu); (2) Haley M. Sapers, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USA (hsapers@caltech.edu); (3) Noah Deutsch, Harvard University Cambridge, Massachusetts, USA (ndeutsch@mde.harvard.edu); (4) Malika Khurana, The New York Times Company, New York, New York, USA (malika.khurana@nytimes.com); (5) John S. Magyar, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (jmagyar@caltech.edu); (6) Sergio A. Parra, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (sparra@caltech.edu); (7) Daniel R. Utter, rwipfler@caltech.edu Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (dutter@caltech.edu); (8) John S. Magyar, Division of Geological and Planetary Sciences, California Institute of Technology Pasadena, California, USA (rwipfler@caltech.edu); (9) David W. Caress, Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (caress@mbari.org); (10) Eric J. Martin Jennifer B. Paduan Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (emartin@mbari.org); (11) Jennifer B. Paduan, Monterey Bay Aquarium Research Institute, Moss Landing, California, USA (paje@mbari.org); (12) Maggie Hendrie, ArtCenter College of Design, Pasadena, California, USA (maggie.hendrie@artcenter.edu); (13) Santiago Lombeyda, California Institute of Technology, Pasadena, California, USA (santiago@caltech.edu); (14) Hillary Mushkin, California Institute of Technology, Pasadena, California, USA (hmushkin@caltech.edu); (15) Alex Endert, Georgia Institute of Technology, Atlanta, Georgia, USA (endert@gatech.edu); (16) Scott Davidoff, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA (scott.davidoff@jpl.nasa.gov); (17) Victoria J. Orphan, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California, USA (vorphan@caltech.edu).

Making Field Research Easier with DeepSee’s Data Visualization

How DeepSee Bridges Gaps in Ocean Research Visualization for Fieldwork Decisions

The Cyber Resilience Act: A Field Guide for CTOs and CISOs

The Evolution of Apache SeaTunnel’s Technical Architecture and Its Applications in the AI Field 

In modern applications, data security is a top priority, especially when sensitive information may be copied or transferred to external applications. By preventing certain actions like "cut," "copy," and "paste" on text fields within iOS apps, we can safeguard text data, ensuring it remains within a trusted group of applications. This article demonstrates how to achieve this protection using Swift, while maintaining full control over text-related operations. Goals of This Approach Restrict External Text Sharing: By managing text actions, we can prevent unauthorized access to sensitive information.
Safe Text Storage and Management: Creating an internal buffer ensures that sensitive text remains protected within the app.
Controlled Clipboard Access: By limiting access to the system clipboard, we can ensure that only authorized applications within a defined group can access this information. Implementation Overview Our approach centers around the TextActionInterceptor class, which "swizzles" standard text-related methods, replacing them with secure alternatives. This class intercepts and overrides the "copy," "paste," and "cut" methods, with an option to enable or disable swizzling dynamically. Core Components Enum for Text Actions: The TextAction enumeration defines all the actions related to text that we want to control.
Saved Method Structure: The SavedTextAction struct stores original implementations of methods, allowing us to restore them as needed.
Swizzling Methods: The toggleSwizzling method manages the swizzling state, enabling or disabling the replacement of standard implementations. Code Example Let’s go through key code segments that bring this functionality to life. class TextActionInterceptor {

    enum TextAction: String {
        case cut
        case copy
        case paste
        case define
        case translate
        case modify
        case update
        case previewItem
    }

    struct SavedTextAction {
        var methodRef: Method
        var originalImplementation: IMP
        var selector: Selector
    }

    private var isSwizzled = false
    var savedActions: [TextAction: SavedTextAction] = [:]
    
    // Method to intercept the "copy" action
    func interceptCopy(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let text = inputField.text(in: range)
        else { return }

        SecureClipboard.shared.contents = text
    }

    // Similar methods for "cut" and "paste"
    func interceptPaste(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let text = SecureClipboard.shared.contents
        else { return }

        inputField.replace(range, withText: text)
    }

    func interceptCut(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let cutText = inputField.text(in: range)
        else { return }

        SecureClipboard.shared.contents = cutText
        inputField.replace(range, withText: "")
    }
    
    // Method to enable or disable swizzling
    func toggleSwizzling(_ enable: Bool) {
        guard enable != isSwizzled else { return }
        isSwizzled = enable
        savedActions.forEach { (_, action) in
            let swizzledMethod = class_getInstanceMethod(
                object_getClass(self),
                action.selector
            )
            guard enable else {
                method_setImplementation(action.methodRef, action.originalImplementation)
                return
            }
            method_exchangeImplementations(action.methodRef, swizzledMethod!)
        }
    }
} Implementing a Secure Clipboard The SecureClipboard class is responsible for managing text stored in the clipboard. It blocks any attempt to share text data externally, ensuring it stays within the app's defined ecosystem. final class SecureClipboard {

    static let restrictedTextPlaceholder = "Access Restricted"
    static var shared = SecureClipboard()
    private let placeholderItem: [String: String]
    private var clipboard: UIPasteboard
    private(set) var internalBuffer: String?

    init() {
        placeholderItem = ["public.utf8-plain-text": SecureClipboard.restrictedTextPlaceholder]
        clipboard = UIPasteboard.general
    }

    var contents: String? {
        get {
            guard
                let clipboardText = clipboard.string,
                clipboardText != SecureClipboard.restrictedTextPlaceholder
            else { return internalBuffer }
            return clipboardText
        }
        set {
            guard let newText = newValue else {
                internalBuffer = nil
                return
            }
            clipboard.items = [placeholderItem]
            internalBuffer = newText.isEmpty ? nil : newText
        }
    }
} How This Works Swizzling of Methods: We replace the standard text action methods with customized, secure methods.
Secure Copying: The interceptCopy method intercepts "copy" actions and transfers the text to the SecureClipboard instead of the system clipboard.
Flexible Management: Swizzling can be dynamically enabled or disabled, allowing for adaptable data security. Summary and Results Applying this approach achieves multiple levels of data security: Access Control: All text operations are managed within the application, preventing potential data leaks.


Safe Data Storage: Text data is accessible only within a controlled application group.


Flexible configuration: The ability to enable and disable swizzling allows flexible control of data security based on conditions. By using swizzling and an internal clipboard, we ensure a high level of data protection in iOS applications. This approach offers a robust solution for maintaining text security across applications in a defined environment. In modern applications, data security is a top priority, especially when sensitive information may be copied or transferred to external applications. By preventing certain actions like "cut," "copy," and "paste" on text fields within iOS apps, we can safeguard text data, ensuring it remains within a trusted group of applications. This article demonstrates how to achieve this protection using Swift, while maintaining full control over text-related operations. Goals of This Approach Restrict External Text Sharing: By managing text actions, we can prevent unauthorized access to sensitive information. Safe Text Storage and Management: Creating an internal buffer ensures that sensitive text remains protected within the app. Controlled Clipboard Access: By limiting access to the system clipboard, we can ensure that only authorized applications within a defined group can access this information. Restrict External Text Sharing : By managing text actions, we can prevent unauthorized access to sensitive information. Restrict External Text Sharing Safe Text Storage and Management : Creating an internal buffer ensures that sensitive text remains protected within the app. Safe Text Storage and Management Controlled Clipboard Access : By limiting access to the system clipboard, we can ensure that only authorized applications within a defined group can access this information. Controlled Clipboard Access Implementation Overview Our approach centers around the TextActionInterceptor class, which "swizzles" standard text-related methods, replacing them with secure alternatives. This class intercepts and overrides the "copy," "paste," and "cut" methods, with an option to enable or disable swizzling dynamically. Core Components Enum for Text Actions: The TextAction enumeration defines all the actions related to text that we want to control. Saved Method Structure: The SavedTextAction struct stores original implementations of methods, allowing us to restore them as needed. Swizzling Methods: The toggleSwizzling method manages the swizzling state, enabling or disabling the replacement of standard implementations. Enum for Text Actions: The TextAction enumeration defines all the actions related to text that we want to control. Enum for Text Actions: The TextAction enumeration defines all the actions related to text that we want to control. Saved Method Structure: The SavedTextAction struct stores original implementations of methods, allowing us to restore them as needed. Saved Method Structure: The SavedTextAction struct stores original implementations of methods, allowing us to restore them as needed. Swizzling Methods: The toggleSwizzling method manages the swizzling state, enabling or disabling the replacement of standard implementations. Swizzling Methods: The toggleSwizzling method manages the swizzling state, enabling or disabling the replacement of standard implementations. Code Example Let’s go through key code segments that bring this functionality to life. class TextActionInterceptor {

    enum TextAction: String {
        case cut
        case copy
        case paste
        case define
        case translate
        case modify
        case update
        case previewItem
    }

    struct SavedTextAction {
        var methodRef: Method
        var originalImplementation: IMP
        var selector: Selector
    }

    private var isSwizzled = false
    var savedActions: [TextAction: SavedTextAction] = [:]
    
    // Method to intercept the "copy" action
    func interceptCopy(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let text = inputField.text(in: range)
        else { return }

        SecureClipboard.shared.contents = text
    }

    // Similar methods for "cut" and "paste"
    func interceptPaste(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let text = SecureClipboard.shared.contents
        else { return }

        inputField.replace(range, withText: text)
    }

    func interceptCut(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let cutText = inputField.text(in: range)
        else { return }

        SecureClipboard.shared.contents = cutText
        inputField.replace(range, withText: "")
    }
    
    // Method to enable or disable swizzling
    func toggleSwizzling(_ enable: Bool) {
        guard enable != isSwizzled else { return }
        isSwizzled = enable
        savedActions.forEach { (_, action) in
            let swizzledMethod = class_getInstanceMethod(
                object_getClass(self),
                action.selector
            )
            guard enable else {
                method_setImplementation(action.methodRef, action.originalImplementation)
                return
            }
            method_exchangeImplementations(action.methodRef, swizzledMethod!)
        }
    }
} class TextActionInterceptor {

    enum TextAction: String {
        case cut
        case copy
        case paste
        case define
        case translate
        case modify
        case update
        case previewItem
    }

    struct SavedTextAction {
        var methodRef: Method
        var originalImplementation: IMP
        var selector: Selector
    }

    private var isSwizzled = false
    var savedActions: [TextAction: SavedTextAction] = [:]
    
    // Method to intercept the "copy" action
    func interceptCopy(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let text = inputField.text(in: range)
        else { return }

        SecureClipboard.shared.contents = text
    }

    // Similar methods for "cut" and "paste"
    func interceptPaste(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let text = SecureClipboard.shared.contents
        else { return }

        inputField.replace(range, withText: text)
    }

    func interceptCut(_ sender: Any?) {
        guard
            let inputField = retrieveTextInput(sender),
            let range = inputField.selectedTextRange,
            let cutText = inputField.text(in: range)
        else { return }

        SecureClipboard.shared.contents = cutText
        inputField.replace(range, withText: "")
    }
    
    // Method to enable or disable swizzling
    func toggleSwizzling(_ enable: Bool) {
        guard enable != isSwizzled else { return }
        isSwizzled = enable
        savedActions.forEach { (_, action) in
            let swizzledMethod = class_getInstanceMethod(
                object_getClass(self),
                action.selector
            )
            guard enable else {
                method_setImplementation(action.methodRef, action.originalImplementation)
                return
            }
            method_exchangeImplementations(action.methodRef, swizzledMethod!)
        }
    }
} Implementing a Secure Clipboard The SecureClipboard class is responsible for managing text stored in the clipboard. It blocks any attempt to share text data externally, ensuring it stays within the app's defined ecosystem. final class SecureClipboard {

    static let restrictedTextPlaceholder = "Access Restricted"
    static var shared = SecureClipboard()
    private let placeholderItem: [String: String]
    private var clipboard: UIPasteboard
    private(set) var internalBuffer: String?

    init() {
        placeholderItem = ["public.utf8-plain-text": SecureClipboard.restrictedTextPlaceholder]
        clipboard = UIPasteboard.general
    }

    var contents: String? {
        get {
            guard
                let clipboardText = clipboard.string,
                clipboardText != SecureClipboard.restrictedTextPlaceholder
            else { return internalBuffer }
            return clipboardText
        }
        set {
            guard let newText = newValue else {
                internalBuffer = nil
                return
            }
            clipboard.items = [placeholderItem]
            internalBuffer = newText.isEmpty ? nil : newText
        }
    }
} final class SecureClipboard {

    static let restrictedTextPlaceholder = "Access Restricted"
    static var shared = SecureClipboard()
    private let placeholderItem: [String: String]
    private var clipboard: UIPasteboard
    private(set) var internalBuffer: String?

    init() {
        placeholderItem = ["public.utf8-plain-text": SecureClipboard.restrictedTextPlaceholder]
        clipboard = UIPasteboard.general
    }

    var contents: String? {
        get {
            guard
                let clipboardText = clipboard.string,
                clipboardText != SecureClipboard.restrictedTextPlaceholder
            else { return internalBuffer }
            return clipboardText
        }
        set {
            guard let newText = newValue else {
                internalBuffer = nil
                return
            }
            clipboard.items = [placeholderItem]
            internalBuffer = newText.isEmpty ? nil : newText
        }
    }
} How This Works Swizzling of Methods: We replace the standard text action methods with customized, secure methods. Secure Copying: The interceptCopy method intercepts "copy" actions and transfers the text to the SecureClipboard instead of the system clipboard. Flexible Management: Swizzling can be dynamically enabled or disabled, allowing for adaptable data security. Swizzling of Methods : We replace the standard text action methods with customized, secure methods. Swizzling of Methods Secure Copying: The interceptCopy method intercepts "copy" actions and transfers the text to the SecureClipboard instead of the system clipboard. Secure Copying: The interceptCopy method intercepts "copy" actions and transfers the text to the SecureClipboard instead of the system clipboard. Flexible Management : Swizzling can be dynamically enabled or disabled, allowing for adaptable data security. Flexible Management Summary and Results Applying this approach achieves multiple levels of data security: Access Control: All text operations are managed within the application, preventing potential data leaks. Safe Data Storage: Text data is accessible only within a controlled application group. Flexible configuration: The ability to enable and disable swizzling allows flexible control of data security based on conditions. Access Control: All text operations are managed within the application, preventing potential data leaks. Access Control : All text operations are managed within the application, preventing potential data leaks. Access Control Safe Data Storage: Text data is accessible only within a controlled application group. Safe Data Storage : Text data is accessible only within a controlled application group. Safe Data Storage Flexible configuration: The ability to enable and disable swizzling allows flexible control of data security based on conditions. Flexible configuration : The ability to enable and disable swizzling allows flexible control of data security based on conditions. Flexible configuration By using swizzling and an internal clipboard, we ensure a high level of data protection in iOS applications. This approach offers a robust solution for maintaining text security across applications in a defined environment.

Securing Text Fields in iOS Apps: Restricting 'Cut,' 'Copy,' and 'Paste' Operations Using Swift

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 B Derivation of Marginal Likelihood with Stochastic Field Amplitude The full signal in time space is given by Using the series representation of the Bessel function, together with Gamma function identities, the 5 random variables can be integrated out analytically. We arrive at the following marginalized (and normalized) likelihood: where This likelihood can be split into three individual likelihoods for the sum/difference peaks and the Compton peak, as given in Eq. (3.2). The form of the likelihoods for the sum and difference peaks is equivalent. 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 B Derivation of Marginal Likelihood with Stochastic Field Amplitude The full signal in time space is given by Using the series representation of the Bessel function, together with Gamma function identities, the 5 random variables can be integrated out analytically. We arrive at the following marginalized (and normalized) likelihood: where This likelihood can be split into three individual likelihoods for the sum/difference peaks and the Compton peak, as given in Eq. (3.2). The form of the likelihoods for the sum and difference peaks is equivalent. 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

Derivation of Marginal Likelihood with Stochastic Field Amplitude

Authors:
(1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing;
(2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing;
(3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 5 Conclusion In conclusion, this paper proposes a novel generalized fictitious play algorithm for computing mixed strategy equilibria in OSMFGs. The key innovations include leveraging an iterative process of solving pure strategy systems to approximate mixed equilibria, as well as expanding the design flexibility for the learning rate parameter. Rigorous convergence results are provided, and finite difference schemes are constructed to efficiently solve the obstacle and Fokker-Planck equations during each iteration. Future work includes extensions to problems with common noise, where the equilibria consist of randomized stopping times that depend on the realized common noise path. The generalized fictitious play framework could also be applied to other competitive games involving optimal stopping decisions. Additionally, further analysis on quantifying the convergence rate and computational complexity could provide deeper theoretical insights. Overall, this paper introduces a novel algorithm and analysis to overcome the limitations of current OSMFG methods, opening the door for handling broader classes of large-scale dynamic games with optimal stopping. Acknowledgement ZZ is supported by the National Key R&D Program of China, Project Number 2021YFA1001200, and the NSFC, grant Number 12031013, 12171013. YL is supported by the NSFC, grant Number 12090022. We thank Xu’an Dou, Jian-Guo Liu and Jiajun Tong for helpful discussions. References [1] Y. Achdou, F. Camilli, and I. C. Dolcetta, Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2012), pp. 2585–2612. [2] ________, Mean field games: Numerical methods for the planning problem, SIAM J. Control. Optim., 50 (2012), pp. 77–109. [3] Y. Achdou and I. C. Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), pp. 1136–1162. [4] R. Aid, R. Dumitrescu, and P. Tankov, The entry and exit game in the electricity markets: A mean-field game approach, Journal of Dynamics & Games, (2020). [5] C. Bertucci, Optimal stopping in mean field games, an obstacle problem approach, Journal de Math´ematiques Pures et Appliqu´ees, (2017). [6] ________, A remark on uzawa’s algorithm and an application to mean field games systems, ESAIM: Mathematical Modelling and Numerical Analysis, (2018). [7] G. Bouveret, R. Dumitrescu, and P. Tankov, Mean-field games of optimal stopping: A relaxed solution approach, SIAM J. Control. Optim., 58 (2018), pp. 1795–1821. [8] L. Brugnano and A. Sestini, Iterative solution of piecewise linear systems for the numerical solution of obstacle problems 12, arXiv: Numerical Analysis, (2009). [9] P. Cardaliaguet and S. Hadikhanloo, Learning in mean field games: The fictitious play, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2015), pp. 569–591. [10] R. Dumitrescu, M. Leutscher, and P. Tankov, Linear programming fictitious play algorithm for mean field games with optimal stopping and absorption, ESAIM: Mathematical Modelling and Numerical Analysis, (2022). [11] X. Fernandez-Real and X. Ros-Oton ´ , Regularity theory for elliptic pde, 2022. [12] R. H. W. Hoppe, Multigrid algorithms for variational inequalities, SIAM Journal on Numerical Analysis, 24 (1987), pp. 1046–1065. [13] J. Huang and T. Xie, A class of mean-field games with optimal stopping and its applications, Asian Journal of Control, (2023). [14] J. M. Lasry and P. L. Lions, Jeux `a champ moyen. i – le cas stationnaire, Comptes Rendus Mathematique, 343 (2006), pp. 619–625. [15] ________, Jeux `a champ moyen. ii – horizon fini et contrˆole optimal, Comptes Rendus Mathematique, 343 (2006), pp. 679–684. [16] ________, Mean field games, Japanese Journal of Mathematics, 2 (2007), pp. 229–260. [17] M. Lauri`ere, Numerical methods for mean field games and mean field type control, ArXiv, abs/2106.06231 (2021). [18] P. Lee, T. W. Kim, and S. Kim, Accurate and efficient numerical solutions for elliptic obstacle problems, Journal of Inequalities and Applications, 2017 (2017). [19] S. Perrin, J. P´erolat, M. Lauri`ere, M. Geist, R. Elie, and O. Pietquin ´ , Fictitious play for mean field games: Continuous time analysis and applications, ArXiv, abs/2007.03458 (2020). [20] G. Tran, H. Schaeffer, W. M. Feldman, and S. Osher, An l1 penalty method for general obstacle problems, SIAM J. Appl. Math., 75 (2014), pp. 1424–1444. [21] X. Yang, G. Wang, and X. Gu, Numerical solution for a parabolic obstacle problem with nonsmooth initial data, Numerical Methods for Partial Differential Equations, 30 (2014). This paper is available on arxiv under CC 4.0 license. Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Authors: Authors: (1) Chengfeng Shen, School of Mathematical Sciences, Peking University, Beijing; (2) Yifan Luo, School of Mathematical Sciences, Peking University, Beijing; (3) Zhennan Zhou, Beijing International Center for Mathematical Research, Peking University. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2 Model and 2.1 Optimal Stopping and Obstacle Problem 2.2 Mean Field Games with Optimal Stopping 2.2 Mean Field Games with Optimal Stopping 2.3 Pure Strategy Equilibrium for OSMFG 2.3 Pure Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 2.4 Mixed Strategy Equilibrium for OSMFG 3 Algorithm Construction and 3.1 Fictitious Play 3 Algorithm Construction and 3.1 Fictitious Play 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.2 Convergence of Fictitious Play to Mixed Strategy Equilibrium 3.3 Algorithm Based on Fictitious Play 3.3 Algorithm Based on Fictitious Play 3.4 Numerical Analysis 3.4 Numerical Analysis 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4 Numerical Experiments and 4.1 A Non-local OSMFG Example 4.2 A Local OSMFG Example 4.2 A Local OSMFG Example 5 Conclusion, Acknowledgement, and References 5 Conclusion, Acknowledgement, and References 5 Conclusion In conclusion, this paper proposes a novel generalized fictitious play algorithm for computing mixed strategy equilibria in OSMFGs. The key innovations include leveraging an iterative process of solving pure strategy systems to approximate mixed equilibria, as well as expanding the design flexibility for the learning rate parameter. Rigorous convergence results are provided, and finite difference schemes are constructed to efficiently solve the obstacle and Fokker-Planck equations during each iteration. Future work includes extensions to problems with common noise, where the equilibria consist of randomized stopping times that depend on the realized common noise path. The generalized fictitious play framework could also be applied to other competitive games involving optimal stopping decisions. Additionally, further analysis on quantifying the convergence rate and computational complexity could provide deeper theoretical insights. Overall, this paper introduces a novel algorithm and analysis to overcome the limitations of current OSMFG methods, opening the door for handling broader classes of large-scale dynamic games with optimal stopping. Acknowledgement ZZ is supported by the National Key R&D Program of China, Project Number 2021YFA1001200, and the NSFC, grant Number 12031013, 12171013. YL is supported by the NSFC, grant Number 12090022. We thank Xu’an Dou, Jian-Guo Liu and Jiajun Tong for helpful discussions. References [1] Y. Achdou, F. Camilli, and I. C. Dolcetta, Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2012), pp. 2585–2612. [2] ________, Mean field games: Numerical methods for the planning problem, SIAM J. Control. Optim., 50 (2012), pp. 77–109. [3] Y. Achdou and I. C. Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), pp. 1136–1162. [4] R. Aid, R. Dumitrescu, and P. Tankov, The entry and exit game in the electricity markets: A mean-field game approach, Journal of Dynamics & Games, (2020). [5] C. Bertucci, Optimal stopping in mean field games, an obstacle problem approach, Journal de Math´ematiques Pures et Appliqu´ees, (2017). [6] ________, A remark on uzawa’s algorithm and an application to mean field games systems, ESAIM: Mathematical Modelling and Numerical Analysis, (2018). [7] G. Bouveret, R. Dumitrescu, and P. Tankov, Mean-field games of optimal stopping: A relaxed solution approach, SIAM J. Control. Optim., 58 (2018), pp. 1795–1821. [8] L. Brugnano and A. Sestini, Iterative solution of piecewise linear systems for the numerical solution of obstacle problems 12, arXiv: Numerical Analysis, (2009). [9] P. Cardaliaguet and S. Hadikhanloo, Learning in mean field games: The fictitious play, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2015), pp. 569–591. [10] R. Dumitrescu, M. Leutscher, and P. Tankov, Linear programming fictitious play algorithm for mean field games with optimal stopping and absorption, ESAIM: Mathematical Modelling and Numerical Analysis, (2022). [11] X. Fernandez-Real and X. Ros-Oton ´ , Regularity theory for elliptic pde, 2022. [12] R. H. W. Hoppe, Multigrid algorithms for variational inequalities, SIAM Journal on Numerical Analysis, 24 (1987), pp. 1046–1065. [13] J. Huang and T. Xie, A class of mean-field games with optimal stopping and its applications, Asian Journal of Control, (2023). [14] J. M. Lasry and P. L. Lions, Jeux `a champ moyen. i – le cas stationnaire, Comptes Rendus Mathematique, 343 (2006), pp. 619–625. [15] ________, Jeux `a champ moyen. ii – horizon fini et contrˆole optimal, Comptes Rendus Mathematique, 343 (2006), pp. 679–684. [16] ________, Mean field games, Japanese Journal of Mathematics, 2 (2007), pp. 229–260. [17] M. Lauri`ere, Numerical methods for mean field games and mean field type control, ArXiv, abs/2106.06231 (2021). [18] P. Lee, T. W. Kim, and S. Kim, Accurate and efficient numerical solutions for elliptic obstacle problems, Journal of Inequalities and Applications, 2017 (2017). [19] S. Perrin, J. P´erolat, M. Lauri`ere, M. Geist, R. Elie, and O. Pietquin ´ , Fictitious play for mean field games: Continuous time analysis and applications, ArXiv, abs/2007.03458 (2020). [20] G. Tran, H. Schaeffer, W. M. Feldman, and S. Osher, An l1 penalty method for general obstacle problems, SIAM J. Appl. Math., 75 (2014), pp. 1424–1444. [21] X. Yang, G. Wang, and X. Gu, Numerical solution for a parabolic obstacle problem with nonsmooth initial data, Numerical Methods for Partial Differential Equations, 30 (2014). This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

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