BabylonAI

babylonai

SAN FRANCISCO, CALIFORNIA - HackerNoon, the technology publishing company, is pleased to announce the nominees for its annual “Startups of the Year” award, featuring the most promising and innovative companies in the San Francisco startup ecosystem. The list, compiled by the HackerNoon team, showcases over 800 startups that are making significant contributions to the San Francisco economy and driving innovation in their respective industries. Nominations for the Startups of the Year 2023 are open! Nominate your company as the . startup of the year in San Fransisco here https://www.youtube.com/watch?v=ZqIW54yMsNc&t=2s&embedable=true Among the nominees in San Francisco for Startups of the Year 2023 are: PoplarML Nodle Cryenx Labs Willow Servicing Retain My Team Remio BabylonAI Fungify Monterey Openlayer Incoggo Polymath Feathery RunWhen Recruiterflow Beacon AI Booth.ai General Micro Technology Enterpret Lemonflow Pixly.tv Hyperstack HireQuotient Found Aviator Finch Dispatch Content Beta Zenpath ScreenMeet All of these startups have demonstrated remarkable growth and innovation in their fields. “We are thrilled to recognize the top startups in San Francisco and celebrate their achievements,” said . “Starting and sustaining a company is difficult. We’re honored to showcase their work and impact.” David Smooke, HackerNoon Founder & CEO is a community-driven voting platform that recognizes startups that are changing technology and the world for the better. Starting May 8th, 2023, the internet can nominate and vote for the best ventures in their city. The voting period will remain open until December 31, 2023, and winners will be announced in January 2024 after a comprehensive evaluation by the HackerNoon team. Startups of the Year 2023 In its , a total of 44,427 startups were nominated in 16 different regions, and the community cast 234,344 votes during a 6-month long campaign in favor of budding companies that were growing and trending across every major city in the world. The received perks from sponsors, like , and a from HackerNoon. previous iteration 1,382 winners 3,000 Brex Points tech company news page As a proponent of the open data movement, HackerNoon releases the unique social data gathered from Startups of the Year on GitHub as “ ” for valuable insights. /where-startups-trend Apart from validation from the world's greenest tech publication, all nominees relevant to their domain, region, and funding. They will also receive an and a free version of their very own on HackerNoon. get free interviews amazing discount package designed for their startup Tech Company News Page These pages include business overviews (such as head count, founding year, bio, social links, and domain ranking) coupled with dynamic updates every time the company is mentioned in HackerNoon stories or relevant headlines around the web. People can , and vote once per day per award for who they think should be Startup of the Year all the way up to December 31st. nominate their favorite startups until June 8th In addition to the award itself, winners receive exclusive interviews, an exclusive NFT from HackerNoon, and a for life. .Tech Domain For more information about the “Startups of the Year 2023” list and the nominated companies, please visit our . FAQ page About HackerNoon HackerNoon is a technology publication that aims to empower developers, entrepreneurs, and tech enthusiasts to share their knowledge and expertise with a global audience. With a readership of over 4M+ monthly visitors, HackerNoon provides a platform for startups and innovators to showcase their work and connect with a vibrant community of like-minded individuals. Companies that blog get 67% more leads - Jumpstart your company’s blogging journey by purchasing our . Startups Package Startups Of The Year 2023 is sponsored by . .Tech Domains Nominations for the Startups of the Year 2023 are open! Nominate your company as the . startup of the year in San Fransisco here

Startups of The Year 2023: 800+ Startups Nominated in San Francisco

Artificial Intelligence (AI) has taken the forefront of conversations from employers across the globe. AI is helping organizations make incredible strides in efficiently handling mundane and repetitive tasks and freeing up valuable time and thought space that employees can use to exercise more creativity, solve complex problems, and otherwise focus on the bigger picture. Let’s take a look at some software companies utilizing AI to create enterprise solutions that will transform the workspace of the future. Twilio is a US-based cloud communications platform acting as a Ucaas service company. Twilio allows software developers to automatically make and receive phone calls, send and receive text messages, and carry out other communication tasks using its web service APIs (Application Programming Interfaces). Twilio About 20 years ago, communications platforms were a complex and expensive task, requiring proprietary hardware from various communications vendors. However, Twilio has enabled its customers to apply machine learning to live voice calls via Media Streams. Media Streams is a new set of APIs which enables developers to layer AI-based functionality, like natural language processing (NLP), keyword detection and sentiment analysis, on voice calls in real-time. This, then, reduces the stress of trying to keep up with customers, forgetting information about them and going through a repetitive task, as AI is known to sieve through data to avoid such. Darktrace is a UK-based, one of the world's leading cyber AI companies and the creator of Autonomous Response technology. Darktrace The Darktrace Cyber AI Analyst is a new technology designed to emulate human thought-patterns to consistently investigate cyber threats at the speed of light. With transformational effects for the security industry, first users of the Cyber AI Analyst reported a 92% decrease in the time required to investigate threats and provide conclusions to authorities. Hence, causing fewer worries and saving time to cover more workspaces. Featurespace is a UK-based provider of adaptive behavioral analytics technology for fraud and risk management in financial services & gaming. Featurespace Featurespace built the ARIC platform to outwit fraud attacks by thinking like each one of their customers. The ARIC platform uses quirk detection to study unusual behavioral data in real-time, spotting abnormalities to block any fraud attack as it happens. At the same time, Featurespace recognizes genuine customers without blocking their activity, therefore, reducing the number of real transactions mistakenly identified as fraud by over 70% and increasing business' profits. Catalyte is a workforce creation technology company and one of the fastest-growing providers of software engineering services in the United States. Catalyte Catalyte recently launched a platform named "Odyssey" which has a ground-breaking ability to transform technology workspaces. Relying on AI-based predictive modeling and enhanced analytics, Odyssey helps multinationals and mid-market enterprises birth high-functioning technology workforces that are distinct, native, cost-efficient and manageable. With the Odyssey platform, companies can now source for, train and deploy top-notch developer talents right under their nose. TopDevz is a US and Canada-based software company that offers other companies simple and complex software development services using elite-level developers, designers, project managers and QA testers in the United States and Canada on a project basis. TopDevz TopDevz is known for its work with global brands like Kraft, Invesco, Bose, Ferrari, W Hotels, TMZ, Urban Outfitters and Williams Sonoma by offering its AI-based hiring technology. In recent times, job openings have outpaced available workers by 17%, making the talent gap really wide. This is due to the frequent launching of new products and services in large scales to provide the needs of today’s consumers. This has caused difficulty for companies to deploy the solutions needed to keep up with demand, especially in the technology world. However, TopDevz has birthed TopDevz Academy which enables prospective talents to go through a testing process to prove that they are qualified for a particular project. This will then, avail potential developers the opportunity to sharpen their skills that need improvement through TopDevz academy before being contracted on a project. Hence, reducing the stress of making their employees go through avoidable stress. Babylon Health is a UK-based health service provider that serves remote consultations with doctors and medical experts via text and video messaging through its mobile application. Babylon Health The company’s approach to healthcare is to combine AI with the medical expertise of humans to deliver flawless access. Babylon's AI has been designed to act as a doctor to provide acquirable healthcare for millions at the click of their phones. It can comprehend and recognize different ways humans express their symptoms. Using this knowledge, added with a patient’s health history and current symptoms, it provides information on probable medical conditions and the quickest approach to it, therefore, reducing medical ignorance, increasing life expectancy and saving medical experts from avoidable stress. Onfido is a software company that aids identity recognition in various businesses using a photo-based identity document, a selfie, and artificial intelligence algorithms. Onfido Onfido makes aspiring employees snap a selfie and a picture of their passport, driver’s license, identity card, or one of 600 other forms of identification. They perform a liveness check by following on-screen instructions involving one of many actions and then, using optical character recognition, facial biometrics, criminal records, and other required records. Onfido processes and gives out a risk report showing the person’s identity status, hence, causing a non-rigorous output in making the workplace a haven. Healx is a UK-based biotechnology company integrating artificial intelligence with elite pharmacology to discover treatments for rare diseases. Healx Healx uses an AI-powered and patient-inspired service called Healnet to accelerates the research and creation of rare diseases treatments. Their AI drug discovery platform uses public and proprietary biomedical data and employs the best knowledge graph for rare diseases. Applying this technology with patient perception and drug discovery expertise, they have been able to discover different treatments with just a 30-member team and save lives. Conclusion The workspace is gradually going through an evolvement and this will, in years to come, increase in a geometrical rate due to the import of artificial intelligence. CEOs make more money, recruiters save energy, employees go through less stress and the masses are enjoying transitions in different facets of life. Hence, the transformation that comes with AI only caused a ripple effect in productivity in the industrial world.

8 Enterprise Software Companies Using Artificial Intelligence to Transform Workspace

Of the two Contemporary Empires of the Babylonians and Medes

A Short History of the World, by H. G. Wells, is part of the HackerNoon Books Series. You can jump to any chapter in this book here . THE LAST BABYLONIAN EMPIRE AND THE EMPIRE OF DARIUS I THE LAST BABYLONIAN EMPIRE AND THE EMPIRE OF DARIUS I We have already mentioned how Assyria became a great military power under Tiglath Pileser III and under the usurper Sargon II. Sargon was not this man’s original name; he adopted it to flatter the conquered Babylonians by reminding them of that ancient founder of the Akkadian Empire, Sargon I, two thousand years before his time. Babylon, for all that it was a conquered city, was of greater population and importance than Nineveh, and its great god Bel Marduk and its traders and priests had to be treated politely. In Mesopotamia in the eighth century B.C. A.D. we are already far beyond the barbaric days when the capture of a town meant loot and massacre. Conquerors sought to propitiate and win the conquered. For a century and a half after Sargon the new Assyrian empire endured and, as we have noted, Assurbanipal (Sardanapalus) held at least lower Egypt. But the power and solidarity of Assyria waned rapidly. Egypt by an effort threw off the foreigner under a Pharoah Psammetichus I, and under Necho II attempted a war of conquest in Syria. By that time Assyria was grappling with foes nearer at hand, and could make but a poor resistance. A Semitic people from south-east Mesopotamia, the Chaldeans, combined with Aryan Medes and Persians from the north-east against Nineveh, and in 606 B.C.—for now we are coming down to exact chronology—took that city. There was a division of the spoils of Assyria. A Median Empire was set up in the north under Cyaxares. It included Nineveh, and its capital was Ecbatana. Eastward it reached to the borders of India. To the south of this in a great crescent was a new Chaldean Empire, the Second Babylonian Empire, which rose to a very great degree of wealth and power under the rule of Nebuchadnezzar the Great (the Nebuchadnezzar of the Bible). The last great days, the greatest days of all, for Babylon began. For a time the two Empires remained at peace, and the daughter of Nebuchadnezzar was married to Cyaxares. Meanwhile Necho II was pursuing his easy conquests in Syria. He had defeated and slain King Josiah of Judah, a small country of which there is more to tell presently, at the battle of Megiddo in 608 B.C., and he pushed on to the Euphrates to encounter not a decadent Assyria but a renascent Babylonia. The Chaldeans dealt very vigorously with the Egyptians. Necho was routed and driven back to Egypt, and the Babylonian frontier pushed down to the ancient Egyptian boundaries. From 606 until 589 B.C. the Second Babylonian Empire flourished insecurely. It flourished so long as it kept the peace with the stronger, hardier Median Empire to the north. And during these sixty-seven years not only life but learning flourished in the ancient city. Even under the Assyrian monarchs and especially under Sardanapalus, Babylon had been a scene of great intellectual activity. Sardanapalus, though an Assyrian, had been quite Babylon-ized. He made a library, a library not of paper but of the clay tablets that were used for writing in Mesopotamia since early Sumerian days. His collection has been unearthed and is perhaps the most precious store of historical material in the world. The last of the Chaldean line of Babylonian monarchs, Nabonidus, had even keener literary tastes. He patronized antiquarian researches, and when a date was worked out by his investigators for the accession of Sargon I he commemorated the fact by inscriptions. But there were many signs of disunion in his empire, and he sought to centralize it by bringing a number of the various local gods to Babylon and setting up temples to them there. This device was to be practised quite successfully by the Romans in later times, but in Babylon it roused the jealousy of the powerful priesthood of Bel Marduk, the dominant god of the Babylonians. They cast about for a possible alternative to Nabonidus and found it in Cyrus the Persian, the ruler of the adjacent Median Empire. Cyrus had already distinguished himself by conquering Croesus, the rich king of Lydia in Eastern Asia Minor. He came up against Babylon, there was a battle outside the walls, and the gates of the city were opened to him (538 B.C.). His soldiers entered the city without fighting. The crown prince Belshazzar, the son of Nabonidus, was feasting, the Bible relates, when a hand appeared and wrote in letters of fire upon the wall these mystical words: “Mene, Mene, Tekel, Upharsin,” which was interpreted by the prophet Daniel, whom he summoned to read the riddle, as “God has numbered thy kingdom and finished it; thou art weighed in the balance and found wanting and thy kingdom is given to the Medes and Persians.” Possibly the priests of Bel Marduk knew something about that writing on the wall. Belshazzar was killed that night, says the Bible. Nabonidus was taken prisoner, and the occupation of the city was so peaceful that the services of Bel Marduk continued without intermission. THE RUINS OF PERSEPOLIS The capital city of the Persian Empire; burnt by Alexander the Great Photo: Major W. F. P. Rodd THE GREAT PORCH OF XERXES, AT PERSEPOLIS Photo: Major W. F. P. Rodd The Persian Empire of Darius I, the first of the new Aryan empires in the seat of the old civilizations, was the greatest empire the world had hitherto seen. It included all Asia Minor and Syria, all the old Assyrian and Babylonian empires, Egypt, the Caucasus and Caspian regions, Media, Persia, and it extended into India as far as the Indus. Such an empire was possible because the horse and rider and the chariot and the made-road had now been brought into the world. Hitherto the ass and ox and the camel for desert use had afforded the swiftest method of transport. Great arterial roads were made by the Persian rulers to hold their new empire, and post horses were always in waiting for the imperial messenger or the traveller with an official permit. Moreover the world was now beginning to use coined money, which greatly facilitated trade and intercourse. But the capital of this vast empire was no longer Babylon. In the long run the priesthood of Bel Marduk gained nothing by their treason. Babylon though still important was now a declining city, and the great cities of the new empire were Persepolis and Susa and Ecbatana. The capital was Susa. Nineveh was already abandoned and sinking into ruins. About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. H. G. Wells (2011). A Short History of the World. Urbana, Illinois: Project Gutenberg. Retrieved October 2022, from https://www.gutenberg.org/files/11870/11870-h/11870-h.htm This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org , located at https://www.gutenberg.org/policy/license.html .

THE LAST BABYLONIAN EMPIRE AND THE EMPIRE OF DARIUS I

The Evolution of Modern Medicine: Chapter I - ASSYRIAN AND BABYLONIAN MEDICINE

Authors:
(1) Agustin Moreno;
(2) Francesco Ruscelli. Table of Links Abstract
Introduction
Preliminaries
The B-signature
GIT sequence: low dimensions
GIT sequence: arbitrary dimension
Appendix A. Stability, the Krein–Moser theorem, and refinements and References Appendix A. Stability, the Krein–Moser theorem, and refinements Now we explain how the GIT sequence topologically encodes the (linear) stability of periodic orbits, and compare it to the basic notions of Krein theory, and the Krein–Moser stability theorem. We will also explain how to obtain Theorem A. We follow the exposition in Ekeland’s book [Eke90] (see also [Ab01]). Consider a linear symplectic ODE One can show that the ODE x˙ = JA(t)x is (strongly) stable if and only if R(T) is (strongly) stable [Eke90]. Moreover, stability is equivalent to R(T) being diagonalizable (i.e. all eigenvalues are semi-simple), with its spectrum lying in the unit circle [Eke90]. Now, consider an elliptic pair {λ, λ} of eigenvalues of a symplectic matrix R. Then any other symplectic matrix close to R will also have simple eigenvalues in the unit circle different from ±1 (otherwise an eigenvalue would have to bifurcate into two, as every eigenvalue comes in quadruples, which is not possible if eigenspaces are 1-dimensional). Therefore in this situation, R is strongly stable. The case of eigenvalues with higher multiplicity is dealt with via Krein theory. Whenever two elliptic eigenvalues come together, this gives a criterion for when they cannot possibly escape the circle and transition into a complex quadruple. This works as follows. if x, y are the corresponding eigenvectors. Moreover, if we consider the generalized eigenspaces Definition A.2. (Krein-positivity/negativity) If λ is an eigenvalue of the symplectic matrix R with |λ| = 1, then the signature (p, q) of Gλ is called the Krein-type or Krein signature of λ. If q = 0, i.e. Gλ is positive definite, λ is said to be Krein-positive. If p = 0, i.e. Gλ is negative definite, λ is said to be Krein-negative. If λ is either Krein-negative or Krein-positive, we say that it is Krein-definite. Otherwise, we say that it is Krein-indefinite. If λ is of Krein-type (p, q), then λ is of Krein-type (q, p) [Eke90]. If λ satisfies |λ| = 1 and it is not semi-simple, then it is easy to show that it is Krein-indefinite [Eke90]. Moreover, ±1 are always Krein-indefinite if they are eigenvalues, as they have real eigenvectors x, which are therefore G-isotropic, i.e. G(x, x) = 0. The following, originally proved by Krein in [Kre1; Kre2; Kre3; Kre4] and independently rediscovered by Moser in [M78], gives a characterization of strong stability in terms of Krein theory: Theorem 3 (Krein–Moser). R is strongly stable if and only if it is stable and all its eigenvalues are Krein-definite. See [Eke90] for a proof. Note that this generalizes the case where all eigenvalues are simple, different from ±1 and in the unit circle, as discussed above. Now, the way that the GIT sequence ties up with Krein theory is the following. Proposition A.1 ([FM]). For a Wonenburger matrix, the Krein signature coincides with the B-signature, for elliptic eigenvalues. Example A.3. As a simple example, to illustrate Proposition A.1, consider the Wonenburger matrices As a corollary of the Krein–Moser theorem and of Proposition A.1, we obtain the following. Theorem 4. Let R be a Wonenburger matrix. Then R is strongly stable if and only if it is stable and all its eigenvalues are B-definite. In higher dimensions, whether or not a given high-multiplicity elliptic or hyperbolic eigenvalue of a Wonenburger matrix can be perturbed to be a complex quadruple is determined by whether or not its B-signature is definite; see e.g. Figure 9 and Remark 5.2. This gives a topological proof of the Krein–Moser theorem in all dimensions, and in fact generalizes it for the hyperbolic case, in the case of Wonenburger matrices, proving Theorem A in the Introduction. References [Ab01] Abbondandolo, Alberto. Morse theory for Hamiltonian systems. Chapman & Hall/CRC Research Notes in Mathematics, 425. Chapman & Hall/CRC, Boca Raton, FL, 2001. xii+189 pp. ISBN: 1-58488- 202-6 [Ay22] Aydin, Cengiz. From Babylonian lunar observations to Floquet multipliers and Conley-Zehnder Indices. Preprint arXiv:2206.07803, 2022. [AFKM] Aydin, Cengiz; Frauenfelder, Urs; Koh, Dayung; Moreno, Agustin. Symplectic methods in space mission design. Proceedings of the 2023 AAS/AIAA Astrodynamics Specialist Conference, 2023. [Br69] Broucke, R. Stability of periodic orbits in the elliptic, restricted three-body problem. AIAA J. 7,1003 (1969). [Eke90] Ekeland, Ivar. Convexity methods in Hamiltonian mechanics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 19. Springer-Verlag, Berlin, 1990. x+247 pp. ISBN: 3-540-50613-6 [FM] Frauenfelder, Urs; Moreno, Agustin. On GIT quotients of the symplectic group, stability and bifurcations of periodic orbits, Journal of Symplectic Geometry. To appear. [FMb] Frauenfelder, Urs; Moreno, Agustin. On doubly symmetric periodic orbits. Celestial Mechanics & Dynamical Astronomy, 135 (2023), no. 2, Paper No. 20. [HD98] Howard, James E.; Dullin, Holger R. Linear stability of natural symplectic maps. Phys. Lett. A 246 (1998), no. 3-4, 273–283. [HM87] Howard, J. E.; MacKay, R. S. Calculation of linear stability boundaries for equilibria of Hamiltonian systems. Phys. Lett. A 122 (1987), no. 6-7, 331–334. Kre1] Krein, M.: Generalization of certain investigations of A.M. Liapunov on linear differential equations with periodic coefficients. Doklady Akad. Nauk USSR 73 (1950) 445-448. Kre2] Krein, M.: On the application of an algebraic proposition in the theory of monodromy matrices. Uspekhi Math. Nauk 6 (1951) 171-177. [Kre3] Krein, M.: On the theory of entire matrix-functions of exponential type. Ukrainian Math. Journal 3 (1951) 164-173. [Kre4] Krein, M.: On some maximum and minimum problems for characteristic numbers and Liapunov stability zones. Prikl. Math. Mekh. 15 (1951) 323-348. [M78] Moser, Jürgen. A fixed point theorem in symplectic geometry. Acta Math. 141 (1978), no. 1-2, 17–34. (A. Moreno) Universität Heidelberg, Mathematisches Institut, Heidelberg, Germany Email address: agustin.moreno2191@gmail.com (F. Ruscelli) Universität Heidelberg, Mathematisches Institut, Heidelberg, Germany Email address: fruscelli@mathi.uni-heidelberg.de This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. Authors: (1) Agustin Moreno; (2) Francesco Ruscelli. Authors: Authors: (1) Agustin Moreno; (2) Francesco Ruscelli. Table of Links Abstract Introduction Preliminaries The B-signature GIT sequence: low dimensions GIT sequence: arbitrary dimension Appendix A. Stability, the Krein–Moser theorem, and refinements and References Abstract Abstract Introduction Introduction Preliminaries Preliminaries The B-signature The B-signature GIT sequence: low dimensions GIT sequence: low dimensions GIT sequence: arbitrary dimension GIT sequence: arbitrary dimension Appendix A. Stability, the Krein–Moser theorem, and refinements and References Appendix A. Stability, the Krein–Moser theorem, and refinements and References Appendix A. Stability, the Krein–Moser theorem, and refinements Now we explain how the GIT sequence topologically encodes the (linear) stability of periodic orbits, and compare it to the basic notions of Krein theory, and the Krein–Moser stability theorem. We will also explain how to obtain Theorem A. We follow the exposition in Ekeland’s book [Eke90] (see also [Ab01]). Consider a linear symplectic ODE One can show that the ODE x˙ = JA(t)x is (strongly) stable if and only if R(T) is (strongly) stable [Eke90]. Moreover, stability is equivalent to R(T) being diagonalizable (i.e. all eigenvalues are semi-simple), with its spectrum lying in the unit circle [Eke90]. Now, consider an elliptic pair {λ, λ} of eigenvalues of a symplectic matrix R. Then any other symplectic matrix close to R will also have simple eigenvalues in the unit circle different from ±1 (otherwise an eigenvalue would have to bifurcate into two, as every eigenvalue comes in quadruples, which is not possible if eigenspaces are 1-dimensional). Therefore in this situation, R is strongly stable. The case of eigenvalues with higher multiplicity is dealt with via Krein theory. Whenever two elliptic eigenvalues come together, this gives a criterion for when they cannot possibly escape the circle and transition into a complex quadruple. This works as follows. if x, y are the corresponding eigenvectors. Moreover, if we consider the generalized eigenspaces Definition A.2. (Krein-positivity/negativity) If λ is an eigenvalue of the symplectic matrix R with |λ| = 1, then the signature (p, q) of Gλ is called the Krein-type or Krein signature of λ. If q = 0, i.e. Gλ is positive definite, λ is said to be Krein-positive. If p = 0, i.e. Gλ is negative definite, λ is said to be Krein-negative. If λ is either Krein-negative or Krein-positive, we say that it is Krein-definite. Otherwise, we say that it is Krein-indefinite. Definition A.2. (Krein-positivity/negativity) If λ is of Krein-type (p, q), then λ is of Krein-type (q, p) [Eke90]. If λ satisfies |λ| = 1 and it is not semi-simple, then it is easy to show that it is Krein-indefinite [Eke90]. Moreover, ±1 are always Krein-indefinite if they are eigenvalues, as they have real eigenvectors x, which are therefore G-isotropic, i.e. G(x, x) = 0. The following, originally proved by Krein in [Kre1; Kre2; Kre3; Kre4] and independently rediscovered by Moser in [M78], gives a characterization of strong stability in terms of Krein theory: Theorem 3 (Krein–Moser). R is strongly stable if and only if it is stable and all its eigenvalues are Krein-definite. Theorem 3 (Krein–Moser). R is strongly stable if and only if it is stable and all its eigenvalues are Krein-definite. See [ Eke90 ] for a proof. Note that this generalizes the case where all eigenvalues are simple, different from ±1 and in the unit circle, as discussed above. Now, the way that the GIT sequence ties up with Krein theory is the following. Eke90 Proposition A.1 ([FM]). For a Wonenburger matrix, the Krein signature coincides with the B-signature, for elliptic eigenvalues. Proposition A.1 ([FM]). For a Wonenburger matrix, the Krein signature coincides with the B-signature, for elliptic eigenvalues. Example A.3. As a simple example, to illustrate Proposition A.1, consider the Wonenburger matrices Example A.3. As a corollary of the Krein–Moser theorem and of Proposition A.1, we obtain the following. Theorem 4. Let R be a Wonenburger matrix. Then R is strongly stable if and only if it is stable and all its eigenvalues are B-definite. Theorem 4. Let R be a Wonenburger matrix. Then R is strongly stable if and only if it is stable and all its eigenvalues are B-definite. In higher dimensions, whether or not a given high-multiplicity elliptic or hyperbolic eigenvalue of a Wonenburger matrix can be perturbed to be a complex quadruple is determined by whether or not its B-signature is definite; see e.g. Figure 9 and Remark 5.2. This gives a topological proof of the Krein–Moser theorem in all dimensions, and in fact generalizes it for the hyperbolic case, in the case of Wonenburger matrices, proving Theorem A in the Introduction. References [Ab01] Abbondandolo, Alberto. Morse theory for Hamiltonian systems. Chapman & Hall/CRC Research Notes in Mathematics, 425. Chapman & Hall/CRC, Boca Raton, FL, 2001. xii+189 pp. ISBN: 1-58488- 202-6 [Ay22] Aydin, Cengiz. From Babylonian lunar observations to Floquet multipliers and Conley-Zehnder Indices. Preprint arXiv:2206.07803, 2022. [AFKM] Aydin, Cengiz; Frauenfelder, Urs; Koh, Dayung; Moreno, Agustin. Symplectic methods in space mission design. Proceedings of the 2023 AAS/AIAA Astrodynamics Specialist Conference, 2023. [Br69] Broucke, R. Stability of periodic orbits in the elliptic, restricted three-body problem. AIAA J. 7,1003 (1969). [Eke90] Ekeland, Ivar. Convexity methods in Hamiltonian mechanics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 19. Springer-Verlag, Berlin, 1990. x+247 pp. ISBN: 3-540-50613-6 [FM] Frauenfelder, Urs; Moreno, Agustin. On GIT quotients of the symplectic group, stability and bifurcations of periodic orbits, Journal of Symplectic Geometry. To appear. [FMb] Frauenfelder, Urs; Moreno, Agustin. On doubly symmetric periodic orbits. Celestial Mechanics & Dynamical Astronomy, 135 (2023), no. 2, Paper No. 20. [HD98] Howard, James E.; Dullin, Holger R. Linear stability of natural symplectic maps. Phys. Lett. A 246 (1998), no. 3-4, 273–283. [HM87] Howard, J. E.; MacKay, R. S. Calculation of linear stability boundaries for equilibria of Hamiltonian systems. Phys. Lett. A 122 (1987), no. 6-7, 331–334. Kre1] Krein, M.: Generalization of certain investigations of A.M. Liapunov on linear differential equations with periodic coefficients. Doklady Akad. Nauk USSR 73 (1950) 445-448. Kre2] Krein, M.: On the application of an algebraic proposition in the theory of monodromy matrices. Uspekhi Math. Nauk 6 (1951) 171-177. [Kre3] Krein, M.: On the theory of entire matrix-functions of exponential type. Ukrainian Math. Journal 3 (1951) 164-173. [Kre4] Krein, M.: On some maximum and minimum problems for characteristic numbers and Liapunov stability zones. Prikl. Math. Mekh. 15 (1951) 323-348. [M78] Moser, Jürgen. A fixed point theorem in symplectic geometry. Acta Math. 141 (1978), no. 1-2, 17–34. (A. Moreno) Universität Heidelberg, Mathematisches Institut, Heidelberg, Germany Email address: agustin.moreno2191@gmail.com (F. Ruscelli) Universität Heidelberg, Mathematisches Institut, Heidelberg, Germany Email address: fruscelli@mathi.uni-heidelberg.de This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. available on arxiv

Appendix A. Stability, the Krein–Moser theorem, and refinements and References

পরিশিষ্ট A. স্থিতিশীলতা, ক্রেইন-মোজার উপপাদ্য, এবং পরিমার্জন এবং উল্লেখ

Anhang A. Stabilität, Krein-Moser-Theorem, Verfeinerungen und Referenzen

Apéndice A. Estabilidad, teorema de Krein-Moser y refinamientos y referencias

Annexe A. Stabilité, théorème de Krein-Moser, affinements et références

परिशिष्ट A. स्थिरता, क्रेन-मोजर प्रमेय, और परिशोधन और संदर्भ

付録A. 安定性、クライン・モーザー定理、改良点と参考文献

부록 A. 안정성, Krein-Moser 정리, 개선 사항 및 참고 자료

Apêndice A. Estabilidade, teorema de Krein-Moser e refinamentos e referências

Приложение A. Устойчивость, теорема Крейна–Мозера, уточнения и литература

Ek A. Kararlılık, Krein-Moser teoremi, iyileştirmeler ve Referanslar

Phụ lục A. Tính ổn định, định lý Krein–Moser, các cải tiến và tài liệu tham khảo

附录 A. 稳定性、Krein–Moser 定理以及改进和参考文献

Four Thousand Years of Blood, Snot, and Tears

চার হাজার বছরের রক্ত, স্নোট এবং অশ্রু

Viertausend Jahre Blut, Rotz und Tränen

Cuatro mil años de sangre, mocos y lágrimas

Quatre mille ans de sang, de morve et de larmes

चार हज़ार साल का खून, बलगम और आँसू

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