Related papers: Data Compression with Prime Numbers
Many emerging computer applications require the processing of large numbers, larger than what a CPU can handle. In fact, the top of the line PCs can only manipulate numbers not longer than 32 bits or 64 bits. This is due to the size of the…
This thesis concerns sequential-access data compression, i.e., by algorithms that read the input one or more times from beginning to end. In one chapter we consider adaptive prefix coding, for which we must read the input character by…
The goal of this thesis is to study the compression problems arising in distributed computing systematically. In the first part of the thesis, we study gradient compression for distributed first-order optimization. We begin by establishing…
Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…
The compression of geometric structures is a relatively new field of data compression. Since about 1995, several articles have dealt with the coding of meshes, using for most of them the following approach: the vertices of the mesh are…
Sketching is a probabilistic data compression technique that has been largely developed in the computer science community. Numerical operations on big datasets can be intolerably slow; sketching algorithms address this issue by generating a…
We are interested in classifying those sets of primes $\mathcal{P}$ such that when we sieve out the integers up to $x$ by the primes in $\mathcal{P}^c$ we are left with roughly the expected number of unsieved integers. In particular, we…
We reveal a relationship between the prime counting function and an operation performed on a unique subsequence of the primes.
The rapid growth of digital data has heightened the demand for efficient lossless compression methods. However, existing algorithms exhibit trade-offs: some achieve high compression ratios, others excel in encoding or decoding speed, and…
We present shuffle coding, a general method for optimal compression of sequences of unordered objects using bits-back coding. Data structures that can be compressed using shuffle coding include multisets, graphs, hypergraphs, and others. We…
We propose basic and natural assumptions under which iterative optimization methods with compressed iterates can be analyzed. This problem is motivated by the practice of federated learning, where a large model stored in the cloud is…
To factor an integer N, given that it is equal to the product of two primes, it suffices to find an integer d satisfying a certain simple numerical test. In this approach, the factorization problem equates to the problem of designing an…
Re-Pair is an effective grammar-based compression scheme achieving strong compression rates in practice. Let $n$, $\sigma$, and $d$ be the text length, alphabet size, and dictionary size of the final grammar, respectively. In their original…
This paper presents a distinctive prime detection approach. This method use GM-(n+1) sequences to effectively eliminate complex numbers. The sequences, which consist of odd a number of (n+1), exclude all components except for the initial…
Nowadays data compressors are applied to many problems of text analysis, but many such applications are developed outside of the framework of mathematical statistics. In this paper we overcome this obstacle and show how several methods of…
Clustering is a widely used technique in data mining applications for discovering patterns in underlying data. Most traditional clustering algorithms are limited to handling datasets that contain either numeric or categorical attributes.…
Data compression is very important feature in terms of saving the memory space. In this proposal, an indexed dictionary based compression is used for text data, where the word's reference in dictionary is used for compression. This approach…
Logarithmic gaps have been used in order to find a periodic component of the sequence of prime numbers, hidden by a random noise (stochastic or chaotic). The recovered period for the sequence of the first 10000 prime numbers is equal to…
We describe a group theoretic analysis of Shor's algorithm and other related hidden subgroup problems in mathematics and relate these to symmetries of molecular and condensed phase assemblies. By recasting Shor's algorithm through the lens…
This article is meant to provide an additional point of view, applying known knowledge, to supply keys that have a series of non-repeating digits, in a manner that is not usually thought of. Traditionally, prime numbers are used in…