Related papers: Prime Factoring and The Complexity Of
In this paper, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea underlying the algorithm is based on…
While it is trivial to multiply two C-finite sequences (just like integers), it is not quite so trivial to "factorize" them, or to decide whether they are "prime". The former is plain linear algebra, while the latter is heavy-duty…
A DFA $\mathcal{A}$ is composite if there exist DFAs $\mathcal{A}_1,\dots,\mathcal{A}_t$ with $\mathcal{L}(\mathcal{A}) = \bigcap_{i=1}^{t} \mathcal{L}(\mathcal{A}_i)$ such that each $\mathcal{A}_i$ has strictly less states than the minimal…
A new set of formulas for primes is presented. These formulas are more efficient and grow much slower than the two known formulas of Mills and Wright. 3 new formulas are explained.
Statistical distribution of the primes in an arithmetic progression is considered. The estimation of prime numbers is given and combinatorial methods are used to calculate the twin primes on the available interval. The distribution and…
The split feasibility problem is to find an element in the intersection of a closed set $C$ and the linear preimage of another closed set $D$, assuming the projections onto $C$ and $D$ are easy to compute. This class of problems arises…
Non-parametric methods avoid the problem of having to specify a particular data generating mechanism, but can be computationally intensive, reducing their accessibility for large data problems. Empirical likelihood, a non-parametric…
The purpose of this article is to delve into the properties of invariants. The properties, explained in [2], reveal new ways to develop algorithms that allow us to test the primality of a number. In this article, some of these are shown,…
Complexity is a multi-faceted phenomenon, involving a variety of features including disorder, nonlinearity, and self-organisation. We use a recently developed rigorous framework for complexity to understand measures of complexity. We…
The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…
A monic polynomial in F_q[t] of degree n over a finite field F_q of odd characteristic can be written as the sum of two irreducible monic elements in F_q[t] of degrees n and n-1 if q is larger than a bound depending only on n. The main tool…
A novel combinatorial formula is developed for for tensor product multiplicities in representation theory. We introduce a difference formula linking these multiplicities to restricted occupancy coefficients via a shifted operator. This…
Composite likelihood has shown promise in settings where the number of parameters $p$ is large due to its ability to break down complex models into simpler components, thus enabling inference even when the full likelihood is not tractable.…
We give an example of a dynamical system which is mixing relative to one of its factors, but for which relative mixing of order three does not hold.
A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^{\alpha}\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number…
The probability that the commutator of two group elements is equal to a given element has been introduced in literature few years ago. Several authors have investigated this notion with methods of the representation theory and with…
New algorithms for prime factorization that outperform the existing ones or take advantage of particular properties of the prime factors can have a practical impact on present implementations of cryptographic algorithms that rely on the…
We propose a simple definition of an explanation for the outcome of a classifier based on concepts from causality. We compare it with previously proposed notions of explanation, and study their complexity. We conduct an experimental…
We discuss the formal aspects of the factorial polynomials and of the associated series. We develop the theory using the formalism of quasi-monomials and prove the usefulness of the method for the solutions of nontrivial difference…
Recently, several researchers have claimed that conclusions obtained from a Bayes factor (or the posterior odds) may contradict those obtained from Bayesian posterior estimation. In this short paper, we wish to point out that no such…