| viXra.org > Data Structures and Algorithms > 2406 |
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| viXra.org > Data Structures and Algorithms > 2406 |
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[4] viXra:2406.0157 [pdf] submitted on 2024-06-26 19:20:20
Authors: Junho Eom
Comments: 16 Pages. 2 figures
At least one prime less than n (n >= 2) is known to be used as a factor for composites between n and n^2, and this is explained by prime wave analysis. In this paper, the prime wave analysis is modified with a modular operator and applied to finding new primes within a limited boundary. In results, using the known primes less than 3, the composites were eliminated, and collected remaining prime candidates within a limited boundary between 3 and 3^2. The boundary was sequentially extended from 3^2 to 9^2, 81^2, and 6561^2 by finding 2, 18, 825, and 2606318 prime candidates; these candidates were verified as new primes using the using online databases. In addition, the boundary was extended from 6561^2 to 43046721^2 and the serial new primes were also found within a randomly selected boundary between 6561^2 and 43046721^2. In general, it was concluded that the prime wave analysis modified with a modular operator could be a practical technique for finding new primes within a limited boundary.
Category: Data Structures and Algorithms
[3] viXra:2406.0088 [pdf] submitted on 2024-06-18 15:59:15
Authors: Bo Tian
Comments: 18 Pages.
In this paper, a new algorithm for solving MEB problem is proposed based on newunderstandings on the geometry property of minimal enclosing ball problem. A substitution ofRitter's algorithm is proposed to get approximate results with higher precision, and a 1+ϵapproximation algorithm is presented to get approximation with specified precision within muchless time comparing with present algorithms.
Category: Data Structures and Algorithms
[2] viXra:2406.0050 [pdf] replaced on 2024-06-27 20:42:09
Authors: Chun-Hu Cui, He-Song Cui
Comments: 32 Pages.
In DeFi (Decentralized Finance) applications, and in dApps (Decentralized Application) generally, it is common to periodically pay interest to users as an incentive, or periodically collect a penalty from them as a deterrent. If we view the penalty as a negative reward, both the interest and penalty problemscome down to the problem of distributing rewards. Reward distribution is quite accomplishable in financial management where general computers are used, but on a blockchain, where computational resources are inherently expensive and the amount of computation per transaction is absolutely limited with a predefined, uniform quota, not only do the system administrators have to pay heavy gas fees if they handle rewards of many users one by one, but the transaction may also be terminated on the way. The computational quota makes it impossible to guarantee processing an unknown number of users. We propose novel algorithms that solve Simple Interest, Simple Burn, Compound Interest, and Compound Burn tasks, which are typical components of DeFi applications. If we put numerical errors aside, these algorithms realize accurate distribution of rewards to an unknown number of users with no approximation, while adhering to the computational quota per transaction. For those who might already be using similar algorithms, we prove the algorithms rigorously so that they can be transparently presented to users. We also introduce reusable concepts and notations in decentralized reasoning, and demonstrate how they can be efficiently used. We demonstrate, through simulated tests spanning over 128 simulated years, that the numerical errors do not grow to a dangerous level.
Category: Data Structures and Algorithms
[1] viXra:2406.0046 [pdf] submitted on 2024-06-10 20:05:12
Authors: Junho Eom
Comments: 16 Pages. 4 figures
Primes less a given number n (n >= 2) determines new primes within a limited area increased with a square (n2) or decreased with a square root (sqrt(��)). As the area is extended, the number of primes is also changed and controlled within an extended area boundary or number boundary, n to n2 or n to sqrt(��). The structure of a number boundary is applied to the Euler product and helps to characterize the Euler’s prime boundary between n and (n2 - 1). The characterized Euler product is used to characterize the non-trivial zeroes derived in an elementary way of Riemann zeta function. Then, the characterized Euler product and non-trivial zeroes are discussed regarding their potential number boundaries. Overall, it is concluded that the characteristic of a number boundary can represent the characteristic of primes, especially the number of primes. As the number boundary is characterized by the increased or decreased exponent while the base or given number n is fixed, it is concluded that the pattern of exponent in the number boundary would be a key to understanding the pattern of primes.
Category: Data Structures and Algorithms
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