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For a given positive integer m, let A = {0, 1,… , m} and q ∈ (m,m+1). A sequence (ci) = c1c2 …. consisting of elements in A is called an expansion of x if ∑ ∞i=1 ciq-i = x. It is known that almost every x belonging to the interval [0,m/(q - 1)] has uncountably many expansions. In this paper we study the existence of expansions (di) of x satisfying the inequalities ∑n i=1 diq-i ⩾ ∑n i=1 ciq-i , n = 1, 2,… , for each expansion (ci) of x. [ABSTRACT FROM AUTHOR]

It is well known that the average position or barycenter of generic orbits for the standard tent map is $ 0.5$. In this paper we prove that for any positive integer $ n$ distinct periodic orbits for the standard tent map with the same barycenter. We also provide some patterns of periodic orbits with the same barycenter. [ABSTRACT FROM AUTHOR]

Let $N(a,m)$ be the least integer $n$ (if it exists) such that $varphi (n)equiv apmod m$. Friedlander and Shparlinski proved that for any $varepsilon >0$ there exists $A=A(varepsilon )>0$ such that for any positive integer $m$ which has no prime divisors $p<(log m)^A$ and any integer $a$ with $gcd (a,m)=1,$ we have the bound $N(a,m)ll m^{3+varepsilon }.$ In the present paper we improve this bound to $N(a,m)ll m^{2+varepsilon }.$ [ABSTRACT FROM AUTHOR]

In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and $83$ powers of $2$. [ABSTRACT FROM AUTHOR]