This paper is a prequel to our recent work, "Equivariant Borel liftings in complex analysis and PDE" (arXiv:2507.12058). While the results presented here were established in that work in a more general and abstract setting, the purpose of this paper is to provide a direct proof of the equivariant Weierstrass theorem. It states that there exists a Borel map assigning to each non-periodic positive divisor $Ī$ an entire function $F_Ī$ such that the divisor of zeroes of $F_Ī$ is $Ī$ and such that $F_{Ī-w}(z) = F_Ī(z+w)$, $w\in\mathbb{C}$. In general, non-periodicity cannot be omitted, and Borel measurability cannot be strengthened to continuity. The two key ingredients are the Runge approximation theorem and the existence of "Borel toasts", which are Borel counterparts of Rokhlin towers from ergodic theory. We do not assume prior knowledge of descriptive set theory and have aimed to make the exposition self-contained, aside from several results taken from graduate textbooks.
We present an elementary proof that the asymptotic entropy of a random walk on a countable abelian group is zero when the entropy of the first step of the random walk is finite. Unlike the traditional proof, our approach does not rely on the boundary theory of random walks. To our best knowledge, our direct proof is new even for the group of integers.
We consider random trigonometric polynomials with general dependent coefficients. We show that under mild hypotheses on the structure of dependence, the asymptotics as the degree goes to infinity of the expected number of real zeros coincides with the independent case. To the best of our knowledge, this universality result is the first obtained in a non-Gaussian dependent context. Our proof highlights the robustness of real zeros, even in the presence of dependencies. These findings bring the behavior of random polynomials closer to real-world models, where dependencies between coefficients are common.
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.
Electronic voting is one of the most interesting application of modern cryptography, as it involves many innovative tools (such as homomorphic public-key encryption, non-interactive zero-knowledge proofs, and distributed cryptography) to guarantee several a priori contradictory security properties: the integrity of the tally and the privacy of the individual votes. While many efficient solutions exist for honest-but-curious voters, that follow the official procedure but try to learn more than just the public result, preventing attacks from malicious voters is much more complex: when voters may have incentive to send biased ballots, the privacy of the ballots is much harder to satisfy, whereas this is the crucial security property for electronic voting. We present a new technique to prove that an ElGamal ciphertext contains a message from a specific subset (quasi-adaptive NIZK of subset membership), using linearly-homomorphic signatures. The proofs are both quite efficient to generate, allowing the use of low-power devices to vote, and randomizable, which is important for the strong receipt-freeness property. They are well-suited to prevent vote-selling and replay attacks, which are the main threats against the privacy in electronic voting, with security proofs in the generic group model and the random oracle model.
Proof-of-Stake (PoS) is often promised to decentralize the blockchain security over Proof-of-Work (PoW) by allowing more people to join without specialized mining hardware. However, there is no consensus in the literature on PoS's centralization, with strong arguments from both sides. Furthermore, theoretical models of PoS often assume very strong conditions that cannot be justified in practice. I relax these assumptions and derive a more realistic model that takes into account trading activity and fees. My model shows that the limiting distribution can be centralized regardless of the initial distribution, reconciling conclusions in prior studies.
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 Ī“1 and Ī“2 respectively. Ghosh and Sarnak call those zeros that lie on the rays 'real' including the arc z=exp (iĪø), Ļ/3 ā¤ Īø ā¤ Ļ/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 Ī“2, and N which are positive integers on Ī“1. On Ī“2 we have C log k zeros if the weight k is sufficiently large and on Ī“1 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 Ī»f (n). Most importantly it requires Deligne's bound Ī»f (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 Ī“1 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 Ī“i , i = 1, 2.
The paper deals with cryptocurrencies and trading. Main goal of this article is to introduce strategy for automated trading on cryptocurrency exchange market. For this purpose we will use algorithm based of Floyd-Warshall algorithm. Article is introductory and can this method can be developed in the future. First, a general introduction to cryptocurrencies is given from the programmer's point of view, some statistics data and figure representing volatility of exchange. Then the article describes some basic strategies for automated trading. Also explained is the algorithm Floyd-Warshall and its modifications for automation arbitrage. An illustrative example is given and a trading algorithm is listed.
Hubert Lacoin, FranƧois Simenhaus, Fabio, Lucio Toninelli
Let \mathcal D be a simply connected, smooth enough domain of \mathbb R^2 . For L>0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on \mathbb Z^2 with initial condition such that \sigma_x=-1 if x\in L\mathcal D and \sigma_x=+1 otherwise. It is conjectured [23] that, in the diffusive limit where space is rescaled by L , time by L^2 and L\to\infty , the boundary of the droplet of " - " spins follows a deterministic anisotropic curve-shortening flow, where the normal velocity at a point of its boundary is given by the local curvature times an explicit function of the local slope. The behavior should be similar at finite temperature T<T_c , with a different temperature-dependent anisotropy function. We prove this conjecture (at zero temperature) when \mathcal D is convex. Existence and regularity of the solution of the deterministic curve-shortening flow is not obvious a priori and is part of our result. To our knowledge, this is the first proof of mean curvature-type droplet shrinking for a model with genuine microscopic dynamics.
It is shown that any free action of a zero-dimensional compact group on the -dimensional Menger compactum is -universal for free actions, and that the orbit space is -classifying. Nonexistence of equivariant mappings between and implies that the orbit space has infinite dimension, where is any compact ANR-space with free action of the group of -adic integers. Knowledge of such nonexistence would then permit proof of the Hilbert-Smith conjecture under the assumption of finite dimensionality for the orbit space.
We study two sets of models: independent percolation models in half spaces Zįµā»Ā¹ x Zā, and Ising/Potts models as well as the Fortuin-Kasteleyn (FK) random cluster models on branching planes T x Z, where Z is the one-dimensional lattice, Zā = {0,1,2,...} and T is a Bethe lattice. We prove that for independent percolation in half spaces, the infinite cluster is unique whenever it exists. For the Ising/Potts models on branching planes, there are (at least) two phase transitions; that is, there exist(s) a unique Gibbs state, tree-like nonunique Gibbs states or plane-like nonunique Gibbs states corresponding to high temperature, intermediate temperature or low temperature. In the low temperature plus phase, the plus infinite cluster is unique and it "traps" the space T x Z and prevents co-existence of the minus infinite cluster. For the FK random cluster models (which are dependent percolation models) on T x Z, the number of infinite (open) clusters may be zero, infinity or one depending on the value of p--the probability of each bond being open. This is an extension of Grimmett and Newman's results for independent percolation on T x Z. We also prove that both the independent percolation model and the FK random cluster models satisfy a finite island property when p is close to 1. Chapter 1 is an introduction. Chapter 2 contains the proof of the uniqueness theorem for independent percolation in half spaces. The proof utilizes only a large deviation estimate and translation invariance of the models along the hyperplane Zįµā»Ā¹ x {0}. The Ising/Potts models and the FK random cluster models on the branching planes are studied in Chapter 3. The methods are to use the FK representation of Ising/Potts systems as dependent percolation models to carry over Grimmett and Newman's results for independent percolation to the Ising/Potts models. However, in order to prove the plane-like behavior of the Ising/Potts models, the corresponding results for independent percolation are not sufficient and this led us to investigate independent percolation again and prove a new finite island property. Chapters 2 and 3 are independent. Readers with basic knowledge of percolation and Ising models can omit chapter 1 and read chapters 2 and 3 directly.
exists for every i (Theorem 9 of [3]).Following Levy the state i is called stable or instantaneous according as gt-as finite or infinite.We refer to [2 ] for the foundations of the theory of Markov chains under consideration.Although knowledge of these foundations will be necessary for a thorough understanding of what follows, we shall strive to make the present paper readable by itself.Let i be a stable state with q,>0; such a state always exists unless P{x(t) =x(0), 0^t<Ā°Ā° }=l [8, p. 375].Suppose that P{x(0)>=*} =1.LetX=X,(w) be the "first sojourn time" in the state i, namely the length of the first tinterval in which x(t, w)=i (seeTheorem 1 of [2]).Then P{\^t} =l-e~">', t^O [3, p. 54].Let j be an arbitrary state (not oo!) and defineIf J9*i, a is the "first entrance time intoj"; ii j = i, a is the "second entrance time into i" (the first being zero by hypothesis).It is easily shown that a is a random variable in the broad sense, namely a measurable w-iunction defined on a measurable w-set whose probability may be less than one.We define its distribution function in the broad sense by Fij(t)=P{a^t}.Now it can be proved that the two random variables X and a-X (which may be zero with positive probability) are independent^).This is a special case of Theorem 5 of [2], but we give a simple proof as follows.Let us first note that the distribution of a -X may be derived as follows.It can be shown that(3) a(w) considered as a point on the i-axis is the limit from the right of points of Sj(w) ={t: x(t, w) =j}; hence we havewhere h = 2~m, ra->ā¢ oo.Now for each 5 > 0 define two random variables X,=X,(w) and a, = aa(w) on the set {w: x(s, w)=i} as follows: X"(w) is the supremum of T such that x(t, w)=i, s^t<s + T; as(w) is the infimum of t such that t>\"(w) and x(t, w) =j.Thus X0 and a0 reduce to the previous X (2) The random variables Zi(w), ā ā ā , z"(w), with domains of definition Ai, ā¢ ā ā¢ , A", are said to be independent iff P{ flLx A*[z*M Set] }/P{ f|Li A*l = LTt-i P{A*tz*W Ā£c*]}/P(At) for every real Cx, ā¢ ā¢ ā¢ , ck.(3) In fact, the set Sj(w) is dense in itself (see [2, Ā§4, (ii)]).