Analytic Erdös-Turán conjectures and Erdös-Fuchs theorem by L. Haddad, C. Helou, J. Pihko

“…The sequences of such coefficients have a lot of similarity with sequences of natural numbers considered in additive number theory. …”
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The chromatic sum of a graph: history and recent developments by Ewa Kubicka

“…The chromatic sum of a graph is the smallest sum of colors among all proper colorings with natural numbers. The strength of a graph is the minimum number of colors necessary to obtain its chromatic sum. …”
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Finding product sets in some classes of amenable groups by Dimitrios Charamaras, Andreas Mountakis

“…In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups and all abelian groups $(G,+)$ with the property that the subgroup $2G := \{g+g : g\in G\}$ has finite index. …”
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On the Recursive Sequence xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj by Stevo Stevic

“…We give a complete picture regarding the behavior of positive solutions of the following important difference equation: xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj, n∈ℕ0, where αi, i∈{1,…,k}, and βj, j∈{1,…,m}, are positive numbers such that ∑i=1kαi=∑j=1mβj=1, and pi, i∈{1,…,k}, and qj, j∈{1,…,m}, are natural numbers such that p1<p2<⋯<pk and q1<q2<⋯<qm. The case when gcd(p1,…,pk,q1,…,qm)=1 is the most important. …”
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Asymptotic Periodicity of a Higher-Order Difference Equation by Stevo Stevic

“…We give a complete picture regarding the asymptotic periodicity of positive solutions of the following difference equation: xn=f(xn−p1,…,xn−pk,xn−q1,…,xn−qm), n∈ℕ0, where pi, i∈{1,…,k}, and qj, j∈{1,…,m}, are natural numbers such that p1<p2<⋯<pk, q1<q2<⋯<qm and gcd(p1,…,pk,q1,…,qm)=1, the function f∈C[(0,∞)k+m, (α,∞)], α>0, is increasing in the first k arguments and decreasing in other m arguments, there is a decreasing function g∈C[(α,∞),(α,∞)] such that g(g(x))=x, x∈(α,∞), x=f(x,…,x︸k,g(x),…,g(x)︸m), x∈(α,∞), limx→α+g(x)=+∞, and limx→+∞g(x)=α. …”
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Common Fixed Points for Asymptotic Pointwise Nonexpansive Mappings in Metric and Banach Spaces by P. Pasom, B. Panyanak

“…We also show that, under some suitable conditions, the sequence {xk}k=1∞ defined by xk+1=(1-tmk)xk⊕tmkTmnky(m-1)k, y(m-1)k=(1-t(m-1)k)xk⊕t(m-1)kTm-1nky(m-2)k,y(m-2)k=(1-t(m-2)k)xk⊕t(m-2)kTm-2nky(m-3)k,…,y2k=(1-t2k)xk⊕t2kT2nky1k,y1k=(1-t1k)xk⊕t1kT1nky0k,y0k=xk,  k∈N, converges to a common fixed point of T1,T2,…,Tm where they are asymptotic pointwise nonexpansive mappings on C, {tik}k=1∞ are sequences in [0,1] for all i=1,2,…,m, and {nk} is an increasing sequence of natural numbers. The related results for uniformly convex Banach spaces are also included.…”
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Algebras with actions and automata by W. Kühnel, M. Pfender, J. Meseguer, I. Sols

“…The particular case of algebras with fixed natural numbers object has been studied by the authors in [23].…”
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Prognostic impact of presenting symptoms of patients with hepatocellular carcinoma by Samuel Jun Ming Lim, Ying Hao, George Boon Bee Goh, Jason Pik Eu Chang, Chee Kiat Tan

“…Introduction: It is not known if the nature, number and duration of presenting symptoms at diagnosis of hepatocellular carcinoma impact on overall survival. …”
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