On some universal sums of generalized polygonal numbers

For $m=3,4,\ldots$ those $p_m(x)=(m-2)x(x-1)/2+x$ with $x\in\mathbb Z$ are called generalized $m$-gonal numbers. Sun [13] studied for what values of positive integers $a,b,c$ the sum $ap_5+bp_5+cp_5$ is universal over $\mathbb Z$ (i.e., any $n\in\mathbb N=\{0,1,2,\ldots\}$ has the form $ap_5(x)+bp_5(y)+cp_5(z)$ with $x,y,z\in\mathbb Z$). We prove that $p_5+bp_5+3p_5\,(b=1,2,3,4,9)$ and $p_5+2p_5+6p_5$ are universal over $\mathbb Z$, as conjectured by Sun. Sun also conjectured that any $n\in\mathbb N$ can be written as $p_3(x)+p_5(y)+p_{11}(z)$ and $3p_3(x)+p_5(y)+p_7(z)$ with $x,y,z\in\mathbb N$; in contrast, we show that $p_3+p_5+p_{11}$ and $3p_3+p_5+p_7$ are universal over $\mathbb Z$. Our proofs are essentially elementary and hence suitable for general readers.Comment: Final published versio

Experimental investigation into amplitude-dependent modal properties of an eleven-span motorway bridge

The authors would like to thank their supporters. New Zealand Earthquake Commission (EQC) Research Foundation provided financial support for experimental work (Grant No. UNI/578). New Zealand Transport Agency (NZTA) provided access to the bridge. Piotr Omenzetter’s work within the LRF Centre for Safety and Reliability Engineering at the University of Aberdeen is supported by Lloyd’s Register Foundation. The Foundation helps to protect life and property by supporting engineering-related education, public engagement and the application of research. Ge-Wei Chen’s doctoral study is supported by China Scholarship Council (CSC) (Grant No. 2011637065).Peer reviewe