It is possible to formulate the polynomial Szemerédi theorem as follows: Let qi(x) ∈ Q[x] with qi(Z) ⊂ Z, 1 ≤ i ≤ k. If E ⊂ N has positive upper density then there are a, n ∈ N such that {a, a+q1(n)−q1(0), a+qk(n)−qk(0)} ⊂ E. Using methods of abstract ergodic theory, topological algebra in βN, and some recently-obtained knowledge concerning the relationship between translations on nilmanifolds ...

A simple q-analogue of the sum of cubes is given. This answers a question posed in this journal by Garrett and Hummel. The sum of cubes and its q-analogues It is well-known that the first n consecutive cubes can be summed in closed form as n ∑ k=1 k = ( n + 1 2 )2 . Recently, Garrett and Hummel discovered the following q-analogue of this result: n ∑ k=1 qk−1 (1− q)(2− qk−1 − q) (1− q)2(1− q2) =...

In the present paper, we introduces the notion of integral type contractive mapping with respect to ordered S-metric space and prove some coupled common fixed point results of integral type contractive mapping in ordered S-metric space. Moreover, we give an example to support our main result.

and Applied Analysis 3 Proposition 1.2. Function σ t is μ-measurable. Proof. Pick a dense set {ri}i 1 in 0,∞ and set Bk { t ∈ T : M ( t, 1 2 rk ) 1 2 M t, rk } , qk t rkχBk t k ∈ N . 1.7 It is easy to see that for all k ∈ N, σ t ≥ qk t μ-a.e on T . Hence, supk≥1qk t ≤ σ t . For μ-a.e t ∈ T , arbitrarily choose ε ∈ 0, σ t . Then, there exists rk ∈ σ t − ε, σ t such that M t, 1/2 rk 1/2 M t, rk ,...

Let $ mathcal{H}(mathbb{D}) $ denote the space of analytic functions on the open unit disc $mathbb{D}$. For a weight $mu$ and a nonnegative integer $n$, the $n$'th weighted type space $ mathcal{W}_mu ^{(n)} $ is the space of all $fin mathcal{H}(mathbb{D}) $ such that $sup_{zin mathbb{D}}mu(z)left|f^{(n)}(z)right|begin{align*}left|f right|_{mathcal{W}_...

many epidemiological studies have found that people living in environments with more green space report better physical and mental health than those with less green space. however, the association between visits to green space and mental health has seldom been studied. the environment surrounding residences and its recreational and commuting opportunities are believed to affect human health and...