Related papers: A new explicit expansion approach to Mersenne prim…
A novel polynomial expansion method of symmetric Boolean functions is described. The method is efficient for symmetric Boolean function with small set of valued numbers and has the linear complexity for elementary symmetric Boolean…
We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong…
We show that the onset of laser multiple filamentation can be described as a critical phenomenon that we characterize both experimentally and numerically by measuring a set of seven critical exponents. This phase transition deviates from…
We study simple Lie algebras generated by extremal elements, over arbitrary fields of arbitrary characteristic. We show: (1) If the extremal geometry contains lines, then the Lie algebra admits a $5 \times 5$-grading that can be…
Recent work by Craig, van Ittersum, and Ono constructs explicit expressions in the partition functions of MacMahon that detect the prime numbers. Furthermore, they define generalizations, the MacMahonesque functions, and prove there are…
In our previous paper, we determined a unified combinatorial framework to look at a large number of colored partition identities, and studied the five identities corresponding to the exceptional modular equations of prime degree of the…
We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form h(n) for natural numbers n, or h(p) with p prime, for appropriate polynomials h with integer coefficients. The…
In this paper we obtain some new estimates for multilinear exponential sums in prime fields with a more general class of weights than previously considered. Our techniques are based on some recent progress of Shkredov on multilinear sums…
We proved the so called complex bounds for multimodal, infinitely renormalizable analytic maps with bounded combinatorics: deep renormalizations have polynomial-like extensions with definite modulus. The complex bounds is the first step to…
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…
Here we demonstrate a sieve for analysing primes and their composites, using equivalence classes based on the modulo 6 return value as applied to the Natural numbers. Five features of this 'Hexile' sieve are reviewed. The first aspect, is…
The parametric complexity is the key quantity in the minimum description length (MDL) approach to statistical model selection. Rissanen and others have shown that the parametric complexity of a statistical model approaches a simple function…
In this paper, we present two classes of Bayesian approaches to the two-sample problem. Our first class of methods extends the Bayesian t-test to include all parametric models in the exponential family and their conjugate priors. Our second…
In the paper, the occurrence of zeros and ones in the binary expansion of the primes is studied. In particular the statement in the title is established. The proof is unconditional.
In the paper, the author elementarily unifies and generalizes eight identities involving the functions $\frac{\pm1}{e^{\pm t}-1}$ and their derivatives. By one of these identities, the author establishes two explicit formulae for computing…
We solve the complex extension of the chiral Gaussian Symplectic Ensemble, defined as a Gaussian two-matrix model of chiral non-Hermitian quaternion real matrices. This leads to the appearance of Laguerre polynomials in the complex plane…
We study an LCM-based analogue of Rowland's GCD-based prime-generating recurrence, introduced by the author in 2008. The multiplicative increments of this sequence are conjectured always to be $1$ or prime, but a complete proof requires a…
Lebesgue's dominated convergence theorem is a crucial pillar of modern analysis, but there are certain areas of the subject where this theorem is deficient. Deeper criteria for convergence of integrals are described in this article.
Permutation methods are commonly used to test significance of regressors of interest in general linear models (GLMs) for functional (image) data sets, in particular for neuroimaging applications as they rely on mild assumptions. Permutation…
We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements…