The article examines the functional characteristics of centralized (CEX) and decentralized (DEX) cryptocurrency exchanges, which play a key role in the operation of the digital economy. The architectural, organizational, and technological aspects of various types of exchanges are analyzed, with their advantages and limitations identified in terms of usability, security, liquidity, regulatory compliance, and availability of trading instruments. It is noted that centralized exchanges offer high order execution speed, extensive functionality, integration with payment infrastructure, and user support. At the same time, they require identity verification, store assets in internal accounts, and are therefore subject to certain risks of centralized control. In contrast, decentralized exchanges operate on the basis of smart contracts, do not store user assets, and do not require KYC procedures, thus ensuring a high level of anonymity. However, DEX platforms are characterized by lower liquidity, a limited range of order types, and a higher requirement for users’ technical competence. The study presents a comparative analysis of trading volumes for key cryptocurrencies on Binance, Bybit, and Coinbase Exchange, as well as on decentralized platforms such as Uniswap, PancakeSwap, and Curve. The results reveal a significant lag in trading volume on DEX compared to centralized platforms. Among the analyzed cryptocurrencies, Ethereum demonstrates the highest daily trading volume on both centralized and decentralized exchanges. The study also focuses on user and asset security on cryptocurrency exchanges. It provides a detailed analysis of the use of two-factor authentication (2FA) mechanisms, customer identification procedures (KYC), and anti-money laundering and counter-terrorist financing (AML/CFT) policies. These tools are an integral part of the infrastructure of centralised exchanges that seek to comply with financial regulations and increase user trust. Based on the conducted analysis, the article outlines the prospects for the development of hybrid exchange models that combine the advantages of centralization and decentralization, and defines directions for further research aimed at enhancing the efficiency, security, and accessibility of digital trading platforms. Keywords: cryptocurrency exchange, centralised exchange, decentralised trading platform, digital security, KYC, DEX, CEX.
Dosun-Fuwari is one of Nikoli’s pencil puzzles. It is known that the generalized Dosun-Fuwari puzzle is NP-complete. Due to the inherent difficulty of the puzzle, solvers may often question whether a solution exists. Such questions highlight the need for a method that can verify the existence of a solution without revealing it, thereby preserving the puzzle’s challenge. In this paper, we propose a physical zero-knowledge proof protocol for the Dosun-Fuwari puzzle, which can be executed using 4mn + 2n cards. Here, m × n is the size of the instance of the puzzle.
Data security during transmission over public networks has become a key concern in an era of rapid digitization. Image data is especially vulnerable since it can be stored or transferred using public cloud services, making it open to illegal access, breaches, and eavesdropping. This work suggests a novel way to integrate blockchain technology with a Chaotic Tent map encryption scheme in order to overcome these issues. The outcome is a Blockchain driven Chaotic Tent Map Encryption Scheme (BCTMES) for secure picture transactions. The idea behind this strategy is to ensure an extra degree of security by fusing the distributed and immutable properties of blockchain technology with the intricate encryption offered by chaotic maps. To ensure that the image is transformed into a cipher form that is resistant to several types of attacks, the proposed BCTMES first encrypts it using the Chaotic Tent map encryption technique. The accompanying signed document is safely kept on the blockchain, and this encrypted image is subsequently uploaded to the cloud. The integrity and authenticity of the image are confirmed upon retrieval by utilizing blockchain's consensus mechanism, adding another layer of security against manipulation. Comprehensive performance evaluations show that BCTMES provides notable enhancements in important security parameters, such as entropy, correlation coefficient, key sensitivity, peak signal-to-noise ratio (PSNR), unified average changing intensity (UACI), and number of pixels change rate (NPCR). In addition to providing good defense against brute-force attacks, the high key size of [Formula: see text] further strengthens the system's resilience. To sum up, the BCTMES effectively addresses a number of prevalent risks to picture security and offers a complete solution that may be implemented in cloud-based settings where data integrity and privacy are crucial. This work suggests a promising path for further investigation and practical uses in secure image transmission.
Open access
Chaos-based Image/Signal Encryption
Advanced Steganography and Watermarking Techniques
Solving Alpha — Version 5.2 The derivation of the fine structure constant, closed from two further directions. The fine structure constant α was derived from the self-reference axiom σ = 1/(1+σ) in the first paper of this series and reaffirmed across Versions 1 through 4. The Pentagon formula α⁻¹ = 360/φ² − 2/φ³ + 1/(3⁵φ⁵) + 1/(7⁷φ⁷) reproduces the Morel 2020 atomic recoil determination of α⁻¹ = 137.035999206(11) to within 0.05σ, with zero free parameters and no experimental input. That derivation stands as originally posted. Version 5.2 does not derive α again. It closes the proof from two further directions, each structurally independent of the original derivation and of each other. The first closure is internal uniqueness. Within a pre-specified coefficient pool drawn from the irreducible representations of the binary icosahedral group, the spectral structure of the 600-cell polytope, and the self-referential reciprocal-power family — defined before the formula is consulted and requiring no knowledge of α — the Pentagon formula is the unique 1σ match to Morel 2020. The nearest structurally distinct competitor sits 139× further from the measured value. The four prime exponents (2, 3, 5, 7) of the formula are independently attested by the seventh spectral moment of the 600-cell adjacency matrix, μ₇ = Tr(A⁷)/1440 = 50,400 = 2⁵ · 3² · 5² · 7. The Pentagon formula is not one of many φ-series that fit; it is the only structurally admissible one. The second closure is external overdetermination. The same number α⁻¹ = 137.036 that the Pentagon formula produces is independently recovered, with no electromagnetic input, from three disconnected non-electromagnetic sectors. The cosmological constant Λ from Planck CMB and BAO, the gravitational coupling G from CODATA torsion balance measurements, and the Hubble expansion rate H₀ from SH0ES distance ladders all sit on a single straight line whose slope is α⁻¹ and whose intercept is φ⁻². The horizontal coordinates of that line are forced by Dirichlet's 1837 class number theorem for the field ℚ(√5). Four disconnected experimental programmes, four independent determinations of α⁻¹, one common value. The original derivation gave the number. The first closure shows that no other formula in the structurally admissible space gives that number. The second closure shows that the same number is the unique slope on which four disconnected experimental sectors agree. The proof was complete in V1; it is now closed on three sides. The asymptotic series for α⁻¹ is presented in fully derived form, with coefficient C_k = 2^(k²) counting the directed coupling configurations among k self-referential modes at maximum entropy equilibrium. The series shares the asymptotic character of QED's own perturbation expansion, with optimal truncation near k = 6 settling within 1.65σ of the most precise measurement. A fifth term is pre-registered before any measurement at the required precision exists to test it. Confirmation of either the Parker 2018 caesium or Fan 2023 electron g−2 determinations as the correct value of α⁻¹ at high significance falsifies the formula at the current truncation order; the framework commits to Morel 2020 as the correct value. The fine structure constant is a theorem of self-referential geometry on the field ℚ(√5). The original derivation, the internal uniqueness closure, and the external overdetermination closure are now on the public record together. Ten revisions between V5 and V5.2 are documented inline; the bone-structure claims survive intact. Supplementary ablation scripts and machine-readable results are deposited alongside this record for full reproducibility. Keywords: fine structure constant, self-reference, 600-cell, binary icosahedral group, Dirichlet class number, asymptotic series, Pentagon Physics, derivation closure, falsifiable prediction, ℚ(√5)
The cryptocurrency ecosystem has witnessed an explosive growth, and this study delves into the mechanisms that fuel this expansion. Focusing on airdrops, staking, farming, and coin burning, the research analyzes a vast dataset of transactions from the Ethereum and Binance Networks. This analysis sheds light on the strategic use of these tools by highlighting transactions that engage users and distribute rewards (e.g., airdrops, staking). Furthermore, the study investigates farming as a method to enhance market efficiency by providing liquidity, and coin burning as a strategy to manage token supply and potentially increase value through scarcity. While effective utilization of these mechanisms can bolster token value and project success, regulatory challenges remain. Ultimately, this study aims to raise public awareness of cryptocurrency transaction types and the associated risks. Although many researchers have studied illicit flows on cryptocurrency transactions, in literature we haven’t confronted with any study regarding transaction types. By analyzing over 107 million transaction records, the research presents the distribution of these transaction types. With the analysis of the transaction types, the definitions of them, the statistical outputs from the more than 107 million transaction records from ERC20 and BEP20 blockchain systems and the analysis of some specific token types such as Pancake Swap and Shiba, this research is the first study to present an academic approach with statistical analysis. This study can provide investors with knowledge that safeguard them from the potential pitfalls of cryptocurrency transactions. Also, this study presents some basic statistics so as to understand main patterns of different types of transactions.
Cryptocurrency staking has become a popular investment option, and multiple cryptocurrency exchanges have been offering staking services to customers. Among the various proof-of-stake cryptocurrencies, Ethereum has become one of the most attractive choices. However, due to the complex architecture of Ethereum, exchanges face several challenges in designing and operating their staking systems while ensuring security and efficiency. In this paper, we analyze the solo-staking method on Ethereum and identify four challenges that exchanges are likely to encounter: wallet configuration, validator key security, stable validator node operation, and profitability. To address these challenges, we propose certain solutions, such as implementing a multi-tiered wallet configuration for customer assets and conducting validator operations on cloud platforms. We have implemented some of these proposed methods on the cloud, and successfully achieved stable operation for the Holesky testnet. We have also identified additional challenges that need to be addressed. We summarize them as open challenges and show the research direction.
This article provides a statistical approach to describe the fit of the most popular cryptocurrencies, building off a previous report, "A Statistical Analysis of Cryptocurrencies." We examined Bitcoin, Ethereum, Tether, Binance, Ripple, Cardano, Solana, and Doge coins. To model our cryptocurrencies, we utilized trading prices between 2017 and 2022 in light of historic events, such as the COVID-19 pandemic. Additionally, we performed a correlation analysis to help understand the relationship between the popular cryptos. Here, we report that the candidate distributions we fit to model the currencies needed to be more independent to describe the return of all popular cryptos. This could be due to the need for Correlation between some of these popular cryptos. We found the generalized hyperbolic and the generalized t showed the best performance of the models tested, though these approaches remained limited in their overall fitness. Their performance also varied by cryptocurrency under investigation, with Tether demonstrating the worst fit across all candidate models. Using our fit models, we also predicted the average daily returns for January 1st, 2023, to February 1st, 2023, and generally found good predictive validity. These results are critical in understanding the movements of cryptos and help better understand the risk associated with trading these currencies.
Despite extensive studies on the secondary market trading of cryptocurrencies, few studies have looked into their listings onto exchanges. This paper thus first presents a case study comparing cryptocurrency listings on the largest and more regulated U.S. exchange, Coinbase, and the largest but less regulated global exchange, Binance. Regarding listing performance, while cryptocurrency listings on both exchanges see significantly positive short-term returns, the more regulated Coinbase sees significantly higher listing returns than the less regulated Binance. Regarding listing choices, while both exchanges tend to list cryptocurrencies with more GitHub development activities, conflicts of interest arise when exchanges list cryptocurrencies that their venture capital arms have previously invested in. Specifically, we find the less regulated Binance more likely to list its self-invested coins with inferior fundamentals, and the apparent agency friction does not seem to be corrected by market forces. To obtain external validity of the lessons learned from the top two exchanges, we further construct an exchange regulation index on a larger sample of 80 qualified exchanges, and confirm the relation between stricter exchange regulations and higher short-term listing returns, controlling for cryptocurrency and exchange attributes.
The goal of this paper is to give a relatively simple proof of some known zero density estimates for Riemann zeta function which are sufficiently strong to break the density hypothesis in a nontrivial part of the critical strip. Apart from a simple but ingenious idea of Halasz the proof uses only classical knowledge about the zeta function, results known since at least hundred years.
Fatma Hachicha, Yosra Ghabri, Khaled Guesmi, Ramzi Benkraiem
Cet article analyse la volatilité des cryptomonnaies à l’aide de la méthode de Monte-Carlo par chaînes de Markov (MCMC). L’objectif de cette étude est de trouver une technique plus efficace pour prévoir et estimer la volatilité, afin de fournir des informations cruciales aux gestionnaires de portefeuille et aux décideurs. Deux modèles ont été examinés : le modèle de volatilité stochastique autorégressive avec distribution t de Student (ARSV-t) et le modèle SVOL, en utilisant l’algorithme de Metropolis Hasting. Les résultats montrent que le modèle ARSV-t est plus performant que le modèle SVOL, surtout lorsqu’on traite des données financières hautement volatiles, telles que celles des cryptomonnaies. De plus, les prévisions obtenues avec le modèle ARSV-t sont plus précises que celles du modèle SVOL. Nous avons également constaté que la signification statistique des variables contrôlant la volatilité stochastique varie en fonction de la période d’estimation (COVID-19, guerre Russie-Ukraine). Ces résultats contribuent à améliorer notre compréhension des prévisions de la volatilité sur le marché des cryptomonnaies.
Sandip Das, A. Lahiri, Soumen Nandi, Sagnik Sen · 5 authors
An $(n,m)$-graph is characterised by having $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to an $(n,m)$-graph $H$, is a vertex mapping that preserves adjacency, direction, and type. The $(n,m)$-chromatic number of $G$, denoted by $χ_{n,m}(G)$, is the minimum value of $|V(H)|$ such that there exists a homomorphism of $G$ to $H$. The theory of homomorphisms of $(n,m)$-graphs have connections with graph theoretic concepts like harmonious coloring, nowhere-zero flows; with other mathematical topics like binary predicate logic, Coxeter groups; and has application to the Query Evaluation Problem (QEP) in graph database. In this article, we show that the arboricity of $G$ is bounded by a function of $χ_{n,m}(G)$ but not the other way around. Additionally, we show that the acyclic chromatic number of $G$ is bounded by a function of $χ_{n,m}(G)$, a result already known in the reverse direction. Furthermore, we prove that the $(n,m)$-chromatic number for the family of graphs with a maximum average degree less than $2+ \frac{2}{4(2n+m)-1}$, including the subfamily of planar graphs with girth at least $8(2n+m)$, equals $2(2n+m)+1$. This improves upon previous findings, which proved the $(n,m)$-chromatic number for planar graphs with girth at least $10(2n+m)-4$ is $2(2n+m)+1$. It is established that the $(n,m)$-chromatic number for the family $\mathcal{T}_2$ of partial $2$-trees is both bounded below and above by quadratic functions of $(2n+m)$, with the lower bound being tight when $(2n+m)=2$. We prove $14 \leq χ_{(0,3)}(\mathcal{T}_2) \leq 15$ and $14 \leq χ_{(1,1)}(\mathcal{T}_2) \leq 21$ which improves both known lower bounds and the former upper bound. Moreover, for the latter upper bound, to the best of our knowledge we provide the first theoretical proof.
In this paper we consider nonnegatively curved finite dimensional Alexandrov spaces with a non-collapsing condition, i.e., such that unit balls have volumes uniformly bounded from below away from zero. We study the relation between the isoperimetric profile, the existence of isoperimetric sets, and the asymptotic structure at infinity of such spaces. In this setting, we prove that the following conditions are equivalent: the space has linear volume growth; it is Gromov--Hausdorff asymptotic to one cylinder at infinity; it has uniformly bounded isoperimetric profile; the entire space is a tubular neighborhood of either a line or a ray. Moreover, on a space satisfying any of the previous conditions, we prove existence of isoperimetric sets for sufficiently large volumes, and we characterize the geometric rigidity at the level of the isoperimetric profile. Specializing our study to the $2$-dimensional case, we prove that unit balls have always volumes uniformly bounded from below away from zero, and we prove existence of isoperimetric sets for every volume, characterizing also their topology when the space has no boundary. The proofs exploit a variational approach, and in particular apply to Riemannian manifolds with nonnegative sectional curvature and to Euclidean convex bodies. Up to the authors' knowledge, most of the results are new even in these smooth cases.
It has been proposed in the literature that the volume of Einstein-Rosen bridge is equal to complexity of state preparation ("Complexity=Volume" conjecture). Taking this statement outside the horizon, one might be tempted to propose "Complexity=Time" correspondence. In this Essay we argue that in a blockchain protocol, which is the foundation of all modern cryptocurrencies, time is emergent and it is defined according to a version of "Complexity=Time".
Bitcoin is one of many crypto currencies used for peer - to - peer transactions accessible to anyone with internet access. It is a decentralized digital currency not backed by any government or other legal entity, making it an attractive alternative to the traditional fiat money system. There is no doubt that crypto currencies are the future of money. However, not all crypto projects will succeed in the long run and some might even turn out to be scams. It?s a jungle out there! How do you know which crypt currency project is going to survive? Our paper can help you identify which projects have a good chance of survival by analyzing their market capitalization trends over time using equicorrelation analysis. This paper examines whether or not Bitcoin returns are dependent on common factors, investigates whether or not Bitcoin returns are i. i. d., tests the efficiency of crypt currency markets, and provides an answer to the following question: are crypto currencies efficient? We'll be looking at crypto currencies and their impact on financial markets. We'll also discuss the challenges of using crypto currencies as a predictor for later price movements and look at equicorrelation and its effect on the crypt currency market. We'll also discuss some of the challenges of using equicorrelation as a predictor for future price movements. Finally, we'll explore some potential applications for equicorrelation within the business world.
Cryptocurrency, without exaggeration, can be called one of the most popular and demanded digital assets. The use of electronic means guarantees anonymity for users. Blockchain technology allows you to trace transactions, but does not provide an opportunity to determine the owner of the wallet. However, the issue of exchanging digital assets for traditional money remains open for many users.\n\nWhat do you need to work with digital currency?\nTo buy cryptocurrency and start working with digital assets, you need a virtual wallet and a Visa or MasterCard bank card. There are the following types of wallets:\n\nLocal. Also called desktop. A local wallet is a program that is installed on a computer.\n\n\nMobile. Application for installation on a mobile device. Programs should be downloaded from the official stores AppStore and Play Market.\n\n\nOnline wallet. It is available from any device, but it is not the safest option, so it is not recommended to use it for storing large amounts. An online wallet is perfect for those who are just getting started with digital assets.\n\n\nHardware. It is a USB device. Suitable for storing large amounts.\n\n\nWhen choosing a wallet, you need to focus on your needs. The most convenient way to pay is from a mobile phone. If you plan to make frequent financial transactions, it is better to use an online wallet. The registration procedure will differ depending on the platform you choose.\n\nOnline exchangers\nExchange services offer the fastest and easiest way to exchange cryptocurrencies. Such sites allow you to withdraw money by various methods, including to a bank card. As a rule, such services operate according to a similar algorithm.\n\nTo exchange funds, you must:\n\n - choose the currency of interest;\n - indicate the method of receiving funds;\n - enter the required information;\n - pass verification.\nThen the exchange service will send the number of the virtual account to which the transaction is made in the specified amount. After the money is transferred, you need to confirm the action. As a rule, the exchange takes no more than 15 minutes, but there are times when the application is processed for more than 2 hours. This is usually due to DDoS attacks on the blockchain platform or digital asset renewal. In this case, it is better to contact the technical support of the service. If the operation does not go through because of your mistake, most likely, the money will be lost.\n\nDepending on the service, the commission can range from 1-2% to 10% and higher. It is necessary to choose a site taking into account the needs and parameters of interest. For this, you can use special aggregators. This will allow you to quickly find the option you are interested in. It is better to choose from services that have been operating for at least three years.\n\nCompany Alligator offers to exchange Bitcoins and other cryptocurrencies on favorable terms. The service has been offering its services since 2015 and is one of the most reliable in Russia, Ukraine and the CIS countries. Millions of users work with Alligator every month. The latest software and servers ensure maximum security for user assets. All user data is securely protected.\n\nThe service offers favorable conditions, a good rate and a small commission. Thanks to qualified technical support, any issues are resolved before they become problems. Transactions usually take several minutes to complete. All transactions are performed quickly, and in total, the exchanger supports more than 150 digital currencies.\n\nP2P sites and their features\nYou can withdraw funds from a cryptocurrency wallet to an electronic or bank account through the P2P platform. There are a lot of such systems, so you can choose the option that will meet your exact requirements. On such platforms, digital assets are exchanged between two users. The P2P platform acts as a guarantor. The commission is set by the service, and in some cases it may be absent altogether.\nThe P2P platform acts as a guarantor of the transaction. The seller is guaranteed to receive money before his digital currency reaches the stranger's account. However, there is one point here: there is no guarantee that the user who buys your digital currency will not use a stolen bank card to transfer funds. To minimize risks, it is recommended to conclude deals with users who have a good reputation. On P2P sites, as a rule, there is a user rating, which determines the percentage of success.\n\nHow to exchange money on the exchange\nMany trading platforms provide for the possibility of withdrawing money to a bank card. This is a convenient and safe way. But many exchanges charge a high commission for direct withdrawals, as trading platforms usually cooperate with counterparties. When transferring funds, the platform first transfers the currency to a third-party service. The exchange partner transfers money to the client, after which the operation can be considered completed.\n\nIn this case, the costs are quite high, but this option is the most transparent and popular. It should be borne in mind that not all exchanges are equally safe. The vulnerability of such services is one of the main problems. Trading platforms attract the attention of hackers, and in the event of a hacked wallet, the probability of a refund is minimal.\n\nIf you decide to make money on digital assets, we recommend that you study several cryptocurrency reviews in order to understand which one is the most promising and which one should work with. It is most profitable to make money on the course races, but for this you need to regularly monitor any fluctuations.
In a fog computing environment, lots of devices need to be authenticated in order to keep the platform being secured. To solve this problem, we turn to blockchain techniques. Unlike the identification cryptographic scheme based on elliptic curves, the proposed 2-adic ring identity authentication scheme inherits the high verification efficiency and high key distribution of sequence ciphers of 2-adic ring theory, and this algorithm adds identity hiding function and trading node supervision function by design. The main designed application scenario of this solution is applicable to the consortium blockchain, and the master nodes are mutually trusting cooperative relations. The node transaction verification and block generation consensus algorithm designed in this solution can be implemented in a set of algorithms, which has higher verification efficiency and easier to be deployed than other solutions. This scheme can be widely used in the fog computing environment.
Open access
advanced mathematical theories
Advanced Steganography and Watermarking Techniques
This chapter reflects on what is called ‘Vedic mathematics’ and its relationship to ancient Indian or ‘true’ Vedic mathematics, i.e. the Śulvasūtras or geometrical treatises of the Vedic literature. It refers especially to Vedic Mathematics , a book written by a former Sankaracharya of Puri, the late Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaj. It is supposed to be based on 16 formulas, in Sanskrit called sūtras , which he allegedly reconstructed from the Atharvaveda. In the following, I argue that ‘Vedic Mathematics’ appears to be a ‘scientifistic’ attempt to make calculative tricks a sort of science sanctioned by the Veda. I demonstrate through examples that these aphorisms nothing to do with the Vedas and are very general. In fact, there is no difference between the laws of Mathematics and Vedic Mathematics. Hence, the term ‘Vedic Mathematics’ is inaccurate. I argue that the epithet of ‘Vedic’ is simply added in order to stake claim to a higher truth. I further argue that the Śulvasūtras were a combination of magico-religious and practical requirements. But this did not exclude the possibility of a theoretical interest, as is evidenced by an elementary relationship between proposition and proof.
One of the crucial cornerstone in cryptocurrency is the ability to generate random sequences and aleatory numbers with a null chance that any eavesdropper could extract information about the one-to-one transactions. Under the assumption that the Quantum Mechanics transition probability is perceived as a fully blockchain operation, random sequences are generated as a tool to protect transactions entirely based on Bitcoins. We test the algorithm with integer-order Bessel functions inside of a full blockchain-based algorithm. Simulations have demonstrated the reliability of a simple model of blockchain in the order of 75.
This dissertation is concerned with the zeros of holomorphic Hecke cusp forms in the space of newforms. We estimate a lower bound for the number of zeros on the imaginary axis and on the vertical line R(z)=1/2 in the upper half plane, both of which are outside the unit circle centered at the origin, and we denote these by δ1 and δ2 respectively. Ghosh and Sarnak call those zeros that lie on the rays 'real' including the arc z=exp (iθ), π/3 ≤ θ ≤ π/2, and they showed that a lower bound for the zeros on those geodesic lines is C log k for all sufficiently large weight k for the level 1 case. We extend their results to the newforms with levels N which are positive integers not divisible by 4 on δ2, and N which are positive integers on δ1. On δ2 we have C log k zeros if the weight k is sufficiently large and on δ1 we assume a nonnegativity result on the first negative Hecke eigenvalue and get a conditional result C log k zeros as the weight k goes to infinity. The analysis is closely related to the knowledge of Hecke eigenvalues λf (n). Most importantly it requires Deligne's bound λf (n) n^e (for every e > 0) with which we look into the proof of Theorem 3.1 in Ghosh and Sarnak cite[1], and get the same the approximation theorem for any level in Chapter 2. The estimation of zeros on δ1 also requires a `good' upper bound for the first negative Hecke eigenvalue for which we investigate an upper bound for central values of Hecke L-functions and a nonnegativity result on those values. Those will be studied in Chapters 3 and 4. In Chapter 5 we estimate lower bounds for the number of zeros on δi , i = 1, 2.
Andrei Yu. Khrennikov, Sergei V. Kozyrev, W. A. Zúñiga-Galindo
Introduction The connections between the Archimedean heat equations with number theory and geometry are well known and deep. Let us mention here the connection with the Riemann zeta function which leads naturally to trace-type formulae, see e.g. [48] and the references therein, and the connection with the Atiyah–Singer index theorem, see e.g. [178] and the references therein. The study of non-Archimedean counterparts of the above-mentioned matters is quite relevant, especially taking into account that the Connes and Deninger programs to attack the Riemann hypothesis lead naturally to these matters, see e.g. [112], [121], [309] and the references therein. For instance, several types of p -adic trace formula have been studied, see e.g. [13], [96], [449] and the references therein. In this chapter we study heat traces and spectral zeta functions attached to certain p -adic Laplacians, denoted as A β , following [105]. Using an approach inspired by the work of Minakshisundaram and Pleijel, see [340]–[342], we find a formula for the trace of the semigroup e − tAβ acting on the space of square integrable functions supported on the unit ball with average zero, see Theorem 12.13. The trace of e − t A β is a p -adic oscillatory integral of Laplace–type. We do not know the exact asymptotics of this integral as t tends to infinity; however, we can obtain a good estimation for its behavior at infinity, see Theorem 12.13 (ii). Several unexpected mathematical situations occur in the p -adic setting. For instance, the spectral zeta functions are p -adic Igusa-type integrals, see Theorem 12.18. The p -adic spectral zeta functions studied here may have infinitely many poles on the boundary of their domain of holomorphy. Thus, to the best of our knowledge, the standard Ikehara Tauberian theorems cannot be applied to obtain the asymptotic behavior for the function encompassing the eigenvalues of A β less than or equal to T ≥ 0. However, we are still able to find good estimates for this function, see Theorem 12.18, Remark 12.19, and Conjecture 12.20. The proofs require several results on certain “boundary-value problems” attached to p -adic heat equations associated with operators A β , see Proposition 12.5, Theorem 12.11, and Proposition 12.12.
This chapter is the first of our applied practice chapters. It walks through the two fundamental Ethereum interactions. In the first project, we will broadcast a transaction to three Ethereum networks. In the second project, we will deploy a simple Hello World contract.
by Harold Diamond and Eira Scourfield Heini Halberstam was born in Brux, Czechoslovakia (today Most, Czech Republic), on 11 September 1926, the only child of Michael and Judita Halberstam. Heini's father had moved to Most from Vienna in the 1920s to become the town's Orthodox Rabbi. When Heini was ten years old, his father died suddenly from a heart attack, and soon after, he and his mother moved to Prague. Following the German invasion of Czechoslovakia, Judita arranged for Heini to study English and, in April 1939, to leave home for England on a Kindertransport train. Heini arrived a week later in London, never to see his mother again. In 1942, she, along with most of Prague's Jews, was deported to a Nazi work camp where she soon died of typhoid. After several placements in England, Heini had the good fortune to come in the care of Anne Welsford who recognized his ability and encouraged and supported him through his university studies. Heini began studying mathematics at University College, London. After completing his degree in two years, graduating about 1947, he began working for a PhD at UCL. He wrote his thesis on analytic number theory under the supervision of Theodor Estermann, and he was awarded his PhD degree in 1952. At that time Klaus Roth was a fellow research student who worked with Estermann and Professor Harold Davenport. Around 1948, Heini was appointed to a lecturing position at the University College of the South West in Exeter. The mathematics department then was small with about eight staff who taught the full syllabus for the External Degree of the University of London; in 1955 the College became the independent University of Exeter. A few months after arriving in Exeter, Heini married his first wife, Heather Peacock. He was subsequently appointed Warden of Crossmead Hall of Residence for men students, a position he held in addition to his lectureship. He and his colleagues Walter Hayman and Paddy Kennedy ran a mini research seminar with the encouragement of the Head of Department, Professor T. Arnold Brown. It was at Exeter that Heini's first paper 1 was published in 1949. Heini spent the academic year 1955–1956 in the United States at Brown University. One of his adventures there was getting a traffic ticket. In later years, Heini was amused to recount the conclusion of the court proceeding, at which the judge pronounced his fine with, ‘Rule Britannia, $5.00 please’. When Heini returned to Exeter in 1956 he undertook the supervision of his first research student, namely the second named author of this section. Like others subsequently, she found him to be an inspiring, challenging, and encouraging supervisor. In 1957 Heini moved to Royal Holloway College, University of London, where he was appointed Reader in Mathematics, and he arranged for Eira to transfer there for the second half of her Master's course and to write her thesis. She benefitted from and much appreciated his strong support throughout her university career and his maintenance of regular academic and personal contact by letter, at conferences and during sabbaticals for the rest of his life. While at Royal Holloway College, Heini regularly attended number theory seminars at UCL, and during this time he began his long involvement in the work of the London Mathematical Society (LMS). In 1962 he was appointed Erasmus Smith's Professor of Mathematics at Trinity College, University of Dublin. Two years later Heini moved to the University of Nottingham, where he served at various times as Head of Department and Dean of the Faculty. Heini and Heather had four children, two of whom live in the United States and two in Britain; Heather was tragically killed in a road accident in 1971. Heini subsequently married Doreen Bramley who has two children, both residing in Britain. They have eight grandchildren. In 1980, Heini came to the Mathematics Department of the University of Illinois in Urbana-Champaign (UIUC). He served as Department Head 1980–1988 and retired as Emeritus Professor in 1996. Heini was held in much esteem, and to mark his retirement, the department held an international conference on number theory in his honor. In spring 2014, another such conference was sponsored in memory of Heini and of Paul and Felice Bateman. During his career, Heini also held visiting positions at Brown, Michigan, UC Berkeley, Syracuse, Ohio State University, Paris, Ulm, Scuola Normale Superiore in Pisa, Tel Aviv, York, Hong Kong and Matscience in Madras (now known as Chennai). Heini was a major figure in number theory whose research ranged over several areas. He first studied probabilistic methods, and his later — and most important — work centered on sieves. Other interests of his were mean value theorems, Waring's problem and combinatorial number theory. Some of his research collaborators were Harold Davenport, Harold Diamond, Peter Elliott, Hans-Egon Richert and Klaus Roth. His conjecture with Elliott on the distribution of primes in arithmetic progressions remains one of the outstanding problems in analytic number theory. Sir William Rowan Hamilton (volume 3) 21 Harold Davenport (four volumes) 43 J. E. Littlewood (volume 2) 49 Loo Keng Hua 50 Recent progress in analytic number theory, Durham, 1979 (proceedings) 48 Analytic number theory, Allerton Park, 1990 (proceedings) 65. One of Heini's particular passions, perhaps remembering how he himself had been aided and encouraged as a child, was promoting talented young people. Heini was an inspiring (if demanding) teacher and mentor. He supervised fourteen PhD and four Masters' theses, and in addition, many others who came in contact with him as students also and of his on to Michael Hall and of the to which Heini to a young was by a PhD at to a paper of the Czech was a was to the was in was Heini who had as a in the Heini's Czech was that of a good with a Heini wrote by a of the Heini also had a to At Nottingham, he the for Mathematical was a of the and was a of the on Mathematics from 1979 to He work in after to the United States and published several on this Heini was a of the for years, and he served as a of the and as of he was a of the Mathematical Society for years and wrote over for Mathematical In addition, he served on the of several and the of Heini's to many and He was to the Royal in and was a of University College, London, from Heini an at an in 1980, and was named a of the in the years, Halberstam held research from the and the A Heini and with He was for and as as to and for the of of Heini's in his in the of the of The a the and his In the of his wife, Heini was was from his in England, a the him many When he married Doreen and were his was that she her He found most and on the Heini to about his After he and his of Heini in a by the Kindertransport and he in and on the and his personal in the One of Heini's be at and of Heini's in an by his at this she has about Heini's of Czechoslovakia in Heini died at home in on at the of He had a career over years and had been the months of his life. Heini was an known figure in number theory, for his work in theory. In addition to his Heini was for his encouraging and and his in people. by Michael When came to the University of Illinois in as a student in 1980, Heini had arrived as of the Mathematics of the of number theory for a student first as the in The from was had and for the It about of the of number theory that this was the only that had In addition to the Paul Harold Diamond, Walter and as as several number then and was also to a position the of four years at the of of had the West as an Recent in number theory that time that and, a that be as a of two primes as a a of two In addition to Paul of methods, to about Heini was an in this arriving on that him to an student, had at home to for who had was in the and had a to with on for the first time was also at the of a degree in to in that the that during much of four years at the of was of good Heini a course in during first year at was Heini's and how the course be Heini was a he wrote and was and as he the While he there were that he that he was in the was with the of how and him for the this and good of the as were to as many later in the at Heini had an and had he at and the about many during Heini's at the of was such a good was on to The during his have attended the number theory seminars regularly and a of One as the seminar Heini to and was a to by then had a was to him to be He At that was a paper on the in a number theory course of where a problem was to for The problem was to that an with the 1 1 in the and student in the course had a combinatorial of the that that was and the the was to a that a of the the number of of the was The paper was on this Heini paper and then the with and was perhaps of the of time he was to to as a department one him to have time for students, this was the His was to and he spent time with to and through of and At was a student, Heini to have the of a he one to along with a of to for When returned the paper with a about an from the a with a in the to was the him this He at with of and — a at the of the then he had of the Heini students to much time working a was of in years, he at the of year that was to After many came from one with a strong one in and one from the University of South Heini was of and him for the in as as the one that at the University of South The had the of the the university position to for both and an was to South and had mathematics he was a at the of the years, Heini in career, also in that a number of returned to Illinois to a Heini was in the he a to see that were to with and as as to career to to and his to his and to time with and He with on a regular on a in Heini a by a of time his and became in that Heini Halberstam was he was at times of a father figure to and a by first of Heini Halberstam from the spring of his course on theory. was a student, in number theory, of the to for the of the had in with the in to Heini's His for the was and his were a of had much in a mathematics course as that about primes and the of about The for the a of Hall and was and appreciated an to One of the in the course has a a and began to Heini's about he to be Paul a seminar on an problem about and began many about a paper by Paul the number of of 1 and problems recognized in this paper many of the which was in Heini's at one was where that half the in research in the of the When to Heini's to him this was that the for that Heini had with the on the to soon was about methods, another that a in in and methods, PhD was in a and was years that returned to The of the thesis in second year of was a that of as the of a a number of to Paul to about the he was an on problems of as of and also of his of the Paul that Heini about and Heini to then to Heini's to the problem with problem to be and in first in the PhD thesis of Heini's fellow student Klaus Roth in Heini himself had studied problems in his PhD thesis he and Roth were supervised by Theodor on to later in his was the of regular with at first through on the which have been much Heini's and later the to Heini as a a in a to a about the and perhaps were on Heini on the of in the United States Britain. He to himself as a He never about his personal life. It was many years later that first that he had been one of the by the first the known for the number of to and was of had how to the the the and Heini to — he to his and as another student of Estermann, also had worked on this problem for his PhD Heini also paper published in the of the he was and to — he had of which he to his students as as to and only for Heini with a in which was that was working to that and about two months was to thesis by the number of for from to that Heini's in a after was the had and there was much for after an of a Heini was in many he and was to on the he course he the with on the had long and and When was a student, was with a problem about to a and Heini he about He and that write to Heini for an that was an Heini have to write him a Heini where was at that was he was on the and A few later a was to about one of and one to which years Heini and wrote only one which in paper had in a seminar Heini was a student, on a of by this was had about how to various worked at for a few to and then returned to thesis had this Heini wrote to in about a paper on along with of his for the mathematics was Heini wrote and the paper as only he have that the was an In after wrote returned to as an Heini was by then and were was for to in Heini for many years, in particular completing his on combinatorial with Harold Diamond and Hans-Egon He to number theory seminar regularly he was to on of Heini was a and and him by Harold Diamond and Doreen and in the of on at the University of Heini was from leave at that time and had a full of he was of his time and to and soon found that Heini was a for after arrived in Nottingham, to Heini with a to about who was at was years old, had never been to and suddenly found himself in a with 50 he He came home from in this he and for the that was served in the him Heini's to this He in his that children, to he soon the had and he be the at home as was served at this During the were in there were many the that to were the was the had been and was The such When in one the and on It was and with Heini had another one that the the and he had the and to on major in perhaps a him in Britain. he with a with Heini from a in theory. He and Hans-Egon of the University of Ulm, on as an in During the course of this which on for many years, Richert and Heini and the with much from several of PhD a and of the on wrote a of this work with student in a published by University When began working on with was about had progress to Heini his he Heini that an was and he was came to working with was to Heini write his long a both and with a of which of wrote a particular He was for years an of the at the University of a that during the for and He was his and him from in and with the in the years after his had many one of by first Heini in his work in which he the to of thesis. had also a of the which he had with Klaus and had of He a seminar in the spring of in at University College London where was a to that was on of theory. was PhD with Estermann, and was on the was in the seminar him Estermann have had a with him soon was to a at that he was a of UCL, He at and how When for the at he and that he his the in for the At the of the Heini a which was only and was to was a first Heini to and along to At have worked has the of department was that he as was he and as he had his an of as an at was a PhD with and Heini had the to was a of that was in the this was in to where the staff in had When arrived in had never on and had only of to an and one has to Heini had a in a of for to in which and which to career, namely for students and a course of on the a about and, after the the of the year in was a in several had attended on the at College in and as a had to an in theory. Heini encouraged He a of on theory which were the of his on the with a good working of and He also encouraged with to work on the and the at that time was one to work he write to which to a Heini to the to of at was a seminar Harold and that Other the year were and over from with and with and during one of Heini encouraged to with about the in a which to this and an on in and that in the Heini and Heini had to he have to that an to was who this have and the came Heini that had to was first international conference and a a of there for the first from and a four with and from about the of At time during the year for and at Heini came to see He for a that was at he have had a with the of department was for an and the one year in was the most and of career was by While in to the to to the one in the spring of was the and he that Paul was a at the in in which was to in number theory at the of Heini at as to be to to and the a from He had who that the support be Heini also on how to the to for was also The was that and both attended the and that was first to the United in the theory of in the theory of of as of a a and a of a of as of a of a a of a and a the of as of and the distribution of the distribution of the distribution of see also on of a at Davenport, see also and The of a at in arithmetic Davenport, see also in arithmetic to the of T. Elliott, published In the there was a good of and much as a of in the and from the of Roth and The of the paper strong to as an for the Heini was of this and a to the as at the The and and in various and in particular to to that had been with by The of the paper had been by of the to the by and for the number of of a number as the of a and two The paper a of the of the for the and as was in The first of Heini's research was with his Roth. work a and of several important in and probabilistic number theory, many of which were only in research The of the and of methods, for theory and work a of on and in number theory. the for on the of a A of A to be the number of of A 1 the A as the of a with a A and the by 1 A and and the of and The be that 1 this as for A and both of to a that this the only that the Other of and of the major in this the probabilistic of Paul and for that One of the that there a such that the with the from and the of A to methods, the of and and the of and Roth. for that in which of the by and for of The that to the few A second with the on which Heini his to this a of the and A a of the number of from a of that in a of The first the was about a years by research with the that of the of 1 1 for of the his that primes that the of the of has been at the of A of the was by a was by and Halberstam In the of the an was by It on the that a be from by the of another for of an only to 1 1 It be to the one has been to to be for a and an for the first of the methods, along with important work of and The for in which had been on an were and by various The was for many and After of was that for of was in The most in was J. to the that number the of a and a number that a of at most two as the was to and by the to a good The was the of Heini's with Richert and later also with Diamond on the of and to to small that where on one The with and the of many which a of then research on for two and was only after It a of and and by of The in the was the the where the of both the and the of the in the of the conference Heini's The work to the A in which was to and the most important in and with this the from were Some of the on known The was supported by work of and William who were The an by for by Halberstam on April with the Kindertransport from number was a of in on the and in to the mother of and of as as the of the have and mother had had to and had in the to she have of by the time the in of the and have judge with and at After this he to his in England and the and of with a in England, and and his career as a He the with a from about the of and a about memory that to on a that by the that The this of one in a of a and an with the of the The of the of the and the the of in to in Nazi in the and also the that in the of In there only a few to the of his mother who him on the Kindertransport in in and of the many she to him for the years through a in she was deported to a camp in His Judita died in a work from and father her only in the he for the first time and her on the there and from there was to her and her number through In a had with father after his him he about his mother that in Prague. he that he of her to her in and a of for her in a — he and his mother had only been moved to from Most by the German the in Czechoslovakia that his mother to Heini on the Kindertransport train. They few in she had a and and was to the had for and she was working on getting her memory of his mother in her in a of that father throughout his — that his mother had him by him to England, a that many Kindertransport he that he had At the mother was and on was the Kindertransport from had about for to mother as the of the and a came over to suddenly that was from on and was mother had and had to on in of life. In never see again. She have in the her for to for two her to were to to write They in the in that father for years, the of the that who from the that others When father the to the a few years he to the in and him was in It was that he the was a that he at the that held him from this The after time him that there was much in the as as were about in for on the of the by the that on the of in to after the of in Heini was by this of the and had a that of during that time of was about life. was in the In after letter, the mother her how he he how his work and in after she he write of course the that father with him for after the — that been from the of the by the of his never Prague. Like a of the he that come to he in to he his a in who had and a of the first Kindertransport He had for of the Kindertransport and to to the and he to to his others had and many others were and father he have to he the of the — the of the in time to this was and to the the was a many of the that in the of that to never be to this to the was as were be for this of the he the of children, for the under which were in the first to as a of the the and the of a that The and father in he later on in his life. He was a and he was and and He never his years to and he himself about the Kindertransport one of his to him that this have been how Heini came to one to about Heini this of and to rest in the as in his that came Heini never his mother his Anne Welsford and was moved by the of both to his and, in the of to his to university and to the career that
The synchronization process inherent to the Bitcoin network gives rise to an infinite-server model with the unusual feature that customers interact. Among the closed-form characteristics that we derive for this model is the busy period distribution which, counterintuitively, does not depend on the arrival rate. We explain this by exploiting the equivalence between two specific service disciplines, which is also used to derive the model's stationary distribution. Next to these closed-form results, the second major contribution concerns an asymptotic result: a fluid limit in the presence of service delays. Since fluid limits arise under scalings of the law-of-large-numbers type, they are usually deterministic, but in the setting of the model discussed in this paper the fluid limit is random (more specifically, of growth-collapse type).