Coinmiles gives you free Bitcoin when you shop online or in-store

Coinmiles

Blockchain technologies have already disrupted a plethora of industries around the world, including retail, food distribution, and supply chains just to name a few. Numerous business leaders developed trust that blockchain can help them with building new financial models and achieving greater efficiency. Yet, although cryptocurrencies tick all of the boxes from the perspective of many businesses and their customers, mass adoption is still out of reach. We talked to Alex Gorshkov, Founder, about solutions for crypto businesses, social unrest and its effects on crypto adoption, building in a bear market, and several other interesting topics. CoinPipe Mining Crisis in Kazakhstan, Geopolitics & Bear Market Kazakhstan (aka a hub for Bitcoin mining) was one of the countries severely gripped by the political crisis at the beginning of the year. The Kazakhstan government banned the use of the Internet, which caused social unrest and made the billion-worth Bitcoin mining industry suffer. Several recent reports indicate that the mining industry has found a way to obtain additional energy, which will, to a certain degree, bring back stability to the mining market, Alex believes. According to him, the vast majority of people have changed their attitude toward cryptocurrency. It is kind of natural to see people looking for safer, more secure, and more reliable alternatives to traditional financial systems, and crypto seems to be one of them. So, instead of thinking that crypto is nothing else but a means for making money, a part of the global population now sees it as a convenient tool for performing transactions. Alex says. “However, we won’t say that doing crypto is not profitable. On the contrary! Everyone who’s been in crypto for more than 5 years knows that the market downtrend is the best opportunity to accumulate more tokens. One of the most popular sayings in crypto goes like this “Be fearful when others are greedy and greedy when others are fearful”. Bear markets are normal trends in crypto, and, more importantly, they are temporary. The ones who are ready to invest now will probably reap numerous benefits once we’re back in the green,” Crypto Regulation Crypto is a largely unregulated industry but hey, that is the entire point of blockchain, right? Regular financial systems can’t provide the freedom crypto can. However, since crypto is a gray area, countries are not trying to resist it but rather strengthen the regulation, introduce cryptocurrency taxes, etc. In Alex's opinion, the answer to whether they can completely control crypto remains to be seen. “Various countries are exploring the possibility of launching CBDCs. There are projects that are working on USDC legislation in the US. We can also see that countries under sanctions are thinking about using crypto for international transactions, which can also contribute to producing a legal framework for cryptocurrencies. *And then, on the other hand, China banned crypto. I’ll just mention the rise of non-custodial wallets out there. Non-custodial wallets can have an unlimited number of users and are even super difficult to regulate,”*Alex explains. According to him, the only way to buy crypto would be via offline machines that are, still, capable of marking addresses. If countries would want to, they’d be able to label certain addresses and limit their access to crypto services. Hopefully, we won’t have many cases like this in the future. Mass Adoption According to Alex, the adoption of the technology by both institutional investors and end users is important, but he still gives a slight advantage to end users. The more people want to start using crypto for regular transactions, the more institutional investors and big players will start paying attention to the market. he says. “We are likely to see mass crypto adoption in the following 5 to 15 years maximum. Crypto will be used for paying for goods and services in cafes, restaurants, stores. We know that the largest payment card processors, Visa and Mastercard are in the process of accepting crypto payments, which is great. The rising number of crypto enthusiasts is even making banks and exchanges consider accepting crypto. Once this happens, that will be the trigger for mass adoption,” El Salvador is one of the use cases Alex presented that proves Bitcoin's legal tender status is more than viable. Somewhere around 20% of businesses in El Salvador accept crypto payments today. Crypto will hopefully become a solvent solution for more countries worldwide in the near future, he predicts. B2B Solutions, NEAR, and CoinPipe turns out to be one of the fiercest competitors to Solana. Huge amounts of money are pumped into the ecosystem and numerous marketplaces and games are built on it. NEAR could technically be one of the most interesting and advanced protocols out there. NEAR Protocol Alex explains. “After all, CoinPipe is also built on NEAR. Our goal is to help businesses of all sizes tap into crypto by simply giving them an easy way to start accepting crypto payments. So many businesses don’t even know if accepting payments in Bitcoin, NEAR, or Ethereum is even legal, which tells us that there is plenty of room for education and getting people interested,” is a finance solution that makes it easy for both offline and online stores and their clients to make crypto transactions. Booming in the online environment is relatively easy and, in this world, crypto will soon become something usual. However, the offline environment poses a couple of challenges. CoinPipe Alex concludes. “Our aim is to make it possible to pay in crypto without requiring the user to convert their crypto to fiat and withdraw the money. One thing is for sure – the market is still forming and the CoinPipe team truly believes that we have the potential to become one the biggest adoption boosters,”

An Insight from CoinPipe's Founder on Crypto Regulation, Mass Adoption, and B2B Solutions for Crypto

Did Craig Wright Fake His Supposed Bitcoin Meeting Notes? 

HDR or SDR? A Study of Scaled and Compressed Videos: Objective Assessment

Comparison of Machine Learning Methods: Discussion and Qualitative Analysis

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References 6. Mirror theorem for toric bundles In this section, we will prove the mirror theorem (Theorem 6.1) for (possibly non-split) toric bundles. Throughout this section, we fix the following data: for any vector bundle V . Using these formulas and the projection formula, it holds that From the calculations above, the right-hand side of (6.2) can be computed as follow: which coincides with (6.3). This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References 6. Mirror theorem for toric bundles In this section, we will prove the mirror theorem (Theorem 6.1) for (possibly non-split) toric bundles. Throughout this section, we fix the following data: for any vector bundle V . Using these formulas and the projection formula, it holds that From the calculations above, the right-hand side of (6.2) can be computed as follow: which coincides with (6.3). 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

A Mirror Theorem for Non-split Toric Bundles: Mirror Theorem for Toric Bundles

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: টরিক বান্ডেলের জন্য মিরর উপপাদ্য

Ein Spiegelsatz für nicht aufgeteilte torische Bündel: Spiegelsatz für torische Bündel

Un teorema del espejo para haces tóricos no divididos: Teorema del espejo para haces tóricos

Un théorème miroir pour les faisceaux toriques non divisés : Théorème miroir pour les faisceaux toriques

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: टॉरिक बंडलों के लिए दर्पण प्रमेय

非分割トーリックバンドルのミラー定理: トーリックバンドルのミラー定理

비분할 토릭 번들에 대한 거울 정리: 토릭 번들에 대한 거울 정리

Um teorema do espelho para pacotes tóricos não divididos: Teorema do espelho para pacotes tóricos

Зеркальная теорема для нерасщепляемых торических расслоений: Зеркальная теорема для торических расслоений

Bölünmemiş Torik Demetler için Ayna Teoremi: Torik Demetler için Ayna Teoremi

Định lý gương cho các gói Toric không phân chia: Định lý gương cho các gói Toric

非分裂环面束的镜像定理：环面束的镜像定理

Author:
(1) Yuki Koto Table of Links Abstract and Intro
Genus-zero Gromov-Witten Theory
Toric Bundles
Lagrangian cones of Toric bundles
Mirror theorem for a product of projectives bundles
Mirror Theorem for Toric Bundles
Appendix A. Equivariant Fourier Transformation and References Abstract We construct an I-function for toric bundles obtained as a fiberwise GIT quotient of a (not necessarily split) vector bundle. This is a generalization of Brown’s I-function for split toric bundles [5] and the I-function for non-split projective bundles [21]. In order to prove the mirror theorem, we establish a characterization of points on the Givental Lagrangian cones of toric bundles and prove a mirror theorem for the twisted Gromov-Witten theory of a fiber product of projective bundles. The former result generalizes Brown’s characterization for split toric bundles [5] to the non-split case. 1. Introduction The genus-zero Gromov-Witten theory of a smooth projective variety X plays a significant role in symplectic geometry, algebraic geometry and mirror symmetry. It can be studied by a mirror theorem [13], that is, by finding a convenient point (called an I-function) on the Givental Lagrangian cone LX [14]. The cone LX is a Lagrangian submanifold of an infinite-dimensional symplectic vector space HX, called the Givental space, and is defined by genus-zero gravitational Gromov-Witten invariants. A mirror theorem for X enables us to compute genus-zero Gromov-Witten invariants of X and study quantum cohomology. This is a generalization of Brown’s result [5, Theorem 2], which gives the same characterization for split toric bundles. There are also similar characterization results for other varieties/stacks; see [8, 23, 11]. This result is a straightforward generalization of the mirror theorem for non-split projective bundles [21, Theorem 3.3]. The key ingredient of the proof is the quantum Riemann-Roch theorem [9, Corollary 4] and the well-known fact [24] that Gromov-Witten invariants of the zero locus of a regular section of a convex vector bundle over a variety X are given by twisted Gromov-Witten invariants of X. The plan of the paper is as follows. In Section 2, we recall the definition of GromovWitten invariants, and introduce the non-equivariant/equivariant/twisted Givental cones and quantum Riemann-Roch theorem. In Section 3, we introduce the notion of split/non-split toric bundles, and summarize the structure of cohomology and the semigroups generated by effective curve classes, which will be needed in the subsequent sections. In Section 4, we establish a characterization theorem (Theorem 4.2) for points on the Lagrangian cone of a toric bundle. In Section 5, we prove a mirror theorem for twisted Gromov-Witten theory of a fiber product of projective bundles over B. In Section 6, we prove the main result (Theorem 6.1) of this paper, that is, a mirror theorem for (possibly non-split) toric bundles. In Appendix A, we briefly explain a Fourier transform of Givental cones, and check that our I-function coincides with the Fourier transform of the I-function of a vector bundle. Acknowledgements. The author is deeply grateful to Hiroshi Iritani for his guidance and enthusiastic support during the writing of this paper. He also would like to thank Yuan-Pin Lee and Fumihiko Sanda for very helpful discussions. This work was supported by JSPS KAKENHI Grant Number 22KJ1717. This paper is available on arxiv under CC 4.0 license. Author: (1) Yuki Koto Author: Author: (1) Yuki Koto Table of Links Abstract and Intro Genus-zero Gromov-Witten Theory Toric Bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Abstract and Intro Abstract and Intro Genus-zero Gromov-Witten Theory Genus-zero Gromov-Witten Theory Toric Bundles Toric Bundles Lagrangian cones of Toric bundles Lagrangian cones of Toric bundles Mirror theorem for a product of projectives bundles Mirror theorem for a product of projectives bundles Mirror Theorem for Toric Bundles Mirror Theorem for Toric Bundles Appendix A. Equivariant Fourier Transformation and References Appendix A. Equivariant Fourier Transformation and References Abstract We construct an I-function for toric bundles obtained as a fiberwise GIT quotient of a (not necessarily split) vector bundle. This is a generalization of Brown’s I-function for split toric bundles [5] and the I-function for non-split projective bundles [21]. In order to prove the mirror theorem, we establish a characterization of points on the Givental Lagrangian cones of toric bundles and prove a mirror theorem for the twisted Gromov-Witten theory of a fiber product of projective bundles. The former result generalizes Brown’s characterization for split toric bundles [5] to the non-split case. 1. Introduction The genus-zero Gromov-Witten theory of a smooth projective variety X plays a significant role in symplectic geometry, algebraic geometry and mirror symmetry. It can be studied by a mirror theorem [13], that is, by finding a convenient point (called an I-function) on the Givental Lagrangian cone LX [14]. The cone LX is a Lagrangian submanifold of an infinite-dimensional symplectic vector space HX, called the Givental space, and is defined by genus-zero gravitational Gromov-Witten invariants. A mirror theorem for X enables us to compute genus-zero Gromov-Witten invariants of X and study quantum cohomology. This is a generalization of Brown’s result [5, Theorem 2], which gives the same characterization for split toric bundles. There are also similar characterization results for other varieties/stacks; see [8, 23, 11]. This result is a straightforward generalization of the mirror theorem for non-split projective bundles [21, Theorem 3.3]. The key ingredient of the proof is the quantum Riemann-Roch theorem [9, Corollary 4] and the well-known fact [24] that Gromov-Witten invariants of the zero locus of a regular section of a convex vector bundle over a variety X are given by twisted Gromov-Witten invariants of X. The plan of the paper is as follows. In Section 2, we recall the definition of GromovWitten invariants, and introduce the non-equivariant/equivariant/twisted Givental cones and quantum Riemann-Roch theorem. In Section 3, we introduce the notion of split/non-split toric bundles, and summarize the structure of cohomology and the semigroups generated by effective curve classes, which will be needed in the subsequent sections. In Section 4, we establish a characterization theorem (Theorem 4.2) for points on the Lagrangian cone of a toric bundle. In Section 5, we prove a mirror theorem for twisted Gromov-Witten theory of a fiber product of projective bundles over B. In Section 6, we prove the main result (Theorem 6.1) of this paper, that is, a mirror theorem for (possibly non-split) toric bundles. In Appendix A, we briefly explain a Fourier transform of Givental cones, and check that our I-function coincides with the Fourier transform of the I-function of a vector bundle. Acknowledgements . The author is deeply grateful to Hiroshi Iritani for his guidance and enthusiastic support during the writing of this paper. He also would like to thank Yuan-Pin Lee and Fumihiko Sanda for very helpful discussions. This work was supported by JSPS KAKENHI Grant Number 22KJ1717. Acknowledgements 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

A Mirror Theorem for Non-split Toric Bundles: Abstract and Intro

অ-বিভক্ত টরিক বান্ডেলের জন্য একটি মিরর উপপাদ্য: বিমূর্ত এবং ভূমিকা

Ein Spiegelsatz für nicht-geteilte torische Bündel: Zusammenfassung und Einführung

Un teorema del espejo para haces tóricos no divididos: resumen e introducción

Un théorème du miroir pour les faisceaux toriques non divisés : résumé et introduction

गैर-विभाजित टॉरिक बंडलों के लिए दर्पण प्रमेय: सार और परिचय

非分割トーリックバンドルのミラー定理: 概要と導入

비분할 토릭 번들을 위한 거울 정리: 요약 및 소개

Um teorema do espelho para pacotes tóricos não divididos: resumo e introdução

Зеркальная теорема для нерасщепимых торических расслоений: Аннотация и введение

Bölünmemiş Torik Paketler için Ayna Teoremi: Özet ve Giriş

Định lý gương cho các gói Toric không phân chia: Tóm tắt và giới thiệu

非分裂环面束的镜像定理：摘要和简介

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

Dabba, a decentralized physical infrastructure network (DePIN), has announced the rollout of 10,000 new wifi hotspots across the globe. This initiative is part of the company's second season of its demand-driven DePIN service, aiming to extend internet access in regions with high data demand, particularly across India. The planned deployment coincides with the advent of a token generation event on the Solana blockchain scheduled for July. This step introduces a novel aspect to Dabba's operational model—tokenization, which is set to facilitate a two-sided marketplace. This marketplace aims to involve both data consumers and providers, incentivizing the latter group for their contributions to building and maintaining robust internet infrastructure. Dabba's approach is grounded in real-world demand for internet services, as evidenced by the deployment of over 1,500 hotspots since the testnet launch two months ago. 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Dabba's initiative is set to change this landscape by leveraging the extensive network of Local Cable Operators (LCOs), which act as micro-ISPs, to deploy hotspots effectively. Furthermore, the experience behind Dabba's team, which includes eight years in network development and a role in Google’s public wifi deployments in India, adds a layer of credibility and insight into the project. Their close cooperation with the Indian government ensures that the necessary legislative framework supports this innovative deployment. As Dabba embarks on this ambitious expansion, the implications for India's connectivity landscape are profound. Not only does this model promise to extend internet access to underserved populations, but it also encapsulates the potential of decentralized solutions in addressing critical infrastructure challenges. By aligning technological innovation with market needs and regulatory frameworks, Dabba's DePIN may well set a precedent for future initiatives aiming to democratize internet access in emerging markets. Don’t forget to like and share the story! Vested Interest Disclosure: This author is an independent contributor publishing via our business blogging program. HackerNoon has reviewed the report for quality, but the claims herein belong to the author. #DYOR. Dabba , a decentralized physical infrastructure network (DePIN), has announced the rollout of 10,000 new wifi hotspots across the globe. This initiative is part of the company's second season of its demand-driven DePIN service, aiming to extend internet access in regions with high data demand, particularly across India. Dabba The planned deployment coincides with the advent of a token generation event on the Solana blockchain scheduled for July. This step introduces a novel aspect to Dabba's operational model—tokenization, which is set to facilitate a two-sided marketplace. This marketplace aims to involve both data consumers and providers, incentivizing the latter group for their contributions to building and maintaining robust internet infrastructure. Dabba's approach is grounded in real-world demand for internet services, as evidenced by the deployment of over 1,500 hotspots since the testnet launch two months ago. This move has connected over 15,000 unique devices, managing over 390 TB of data usage. The introduction of the DePIN model not only meets an existing market need but also presents a sustainable business model poised for scaling, especially in high-demand areas. Dabba 's performance over the last two months, showing the deployment of 1,500 hotspots, connection of 15,000 unique devices, consumption of 390 TB of data, and generation of $4.5K in revenue. These metrics highlight the effective expansion and user engagement within Dabba's network, indicating strong demand and initial financial viability as they prepare to scale up operations with 10,000 more hotspots. Dabba The rollout targets India, a nation with a vast number of internet users yet plagued by significant access disparities. Despite being the world's second-largest internet user base, nearly 44% of Indian residents lack adequate internet access, particularly in rural and semi-urban areas. Dabba's initiative is set to change this landscape by leveraging the extensive network of Local Cable Operators (LCOs), which act as micro-ISPs, to deploy hotspots effectively. Furthermore, the experience behind Dabba's team, which includes eight years in network development and a role in Google’s public wifi deployments in India, adds a layer of credibility and insight into the project. Their close cooperation with the Indian government ensures that the necessary legislative framework supports this innovative deployment. As Dabba embarks on this ambitious expansion, the implications for India's connectivity landscape are profound. Not only does this model promise to extend internet access to underserved populations, but it also encapsulates the potential of decentralized solutions in addressing critical infrastructure challenges. By aligning technological innovation with market needs and regulatory frameworks, Dabba's DePIN may well set a precedent for future initiatives aiming to democratize internet access in emerging markets. Don’t forget to like and share the story! Vested Interest Disclosure: This author is an independent contributor publishing via our business blogging program. HackerNoon has reviewed the report for quality, but the claims herein belong to the author. #DYOR. Vested Interest Disclosure: This author is an independent contributor publishing via our business blogging program . HackerNoon has reviewed the report for quality, but the claims herein belong to the author. #DYOR. Vested Interest Disclosure: business blogging program business blogging program

Dabba to Setup 10,000 New Hotspots to Enhance Internet Connectivity in India 

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Dabba richtet 10.000 neue Hotspots ein, um die Internetverbindung in Indien zu verbessern

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ダバ、インドのインターネット接続を強化するため1万箇所のホットスポットを設置

Dabba, 인도에서 인터넷 연결을 향상시키기 위해 10,000개의 새로운 핫스팟 설치 

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Dabba thiết lập 10.000 điểm truy cập mới để tăng cường kết nối Internet ở Ấn Độ 

Dabba 将建立 10,000 个新热点以增强印度互联网连接

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