Pairs of additive forms and Artin's conjecture
Pairs of additive quadratic forms modulo one
Pairs of Bilinear and Quadratic Equations in a Finite Field.
Pairs of cubic forms in many variables
Pairs of Pell equations having at most one common solution in positive integers.
Pairs of polynomials small at a number to certain algebraic powers
Pairs of Quadratic Forms.
Pairs of square-free values of the type ,
We show that there exist infinitely many consecutive square-free numbers of the form , . We also establish an asymptotic formula for the number of such square-free pairs when does not exceed given sufficiently large positive number.
Palindromes in different bases: a conjecture of J. Ernest Wilkins.
Palindromic complexity of infinite words associated with non-simple Parry numbers
We study the palindromic complexity of infinite words , the fixed points of the substitution over a binary alphabet, , , with , which are canonically associated with quadratic non-simple Parry numbers .
Palindromic complexity of infinite words associated with non-simple Parry numbers
We study the palindromic complexity of infinite words uβ, the fixed points of the substitution over a binary alphabet, φ(0) = 0a1, φ(1) = 0b1, with a - 1 ≥ b ≥ 1, which are canonically associated with quadratic non-simple Parry numbers β.
Palindromic complexity of infinite words associated with simple Parry numbers
A simple Parry number is a real number such that the Rényi expansion of is finite, of the form . We study the palindromic structure of infinite aperiodic words that are the fixed point of a substitution associated with a simple Parry number . It is shown that the word contains infinitely many palindromes if and only if . Numbers satisfying this condition are the so-called confluent Pisot numbers. If then is an Arnoux-Rauzy word. We show that if is a confluent Pisot number then...
Palindromic continued fractions
In the present work, we investigate real numbers whose sequence of partial quotients enjoys some combinatorial properties involving the notion of palindrome. We provide three new transendence criteria, that apply to a broad class of continued fraction expansions, including expansions with unbounded partial quotients. Their proofs heavily depend on the Schmidt Subspace Theorem.
Palindromic powers.
Palindromic squares for various number system bases
Parabolic, hyperbolic and elliptic Poincaré series
Parallelepipeds, nilpotent groups and Gowers norms
In his proof of Szemerédi’s Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions and and show when this possible, and describe a correspondence between the parallelepiped structures and nilpotent groups.
Parallélogrammes galoisiens infinis
Parameter spaces for quadrics
The parameter spaces for quadrics are reviewed. In addition, an explicit formula for the number of quadrics tangent to given linear subspaces is presented.
Parameterized families of quadratic number fields with 3-rank at least 2