No abstract

We show the existence of unbounded solutions to difference equations of the form { x n + 1 = c ′ n x n B n y n , y n + 1 = b n x n + c n y n A n + C n y n           f o r       n = 0 , 1 , … , \left\{ {\matrix{{{x_{n + 1}} = {{{{c'}_n}{x_n}} \over {{B_n}{y_n}}},} \hfill \cr {{y_{n + 1}} = {{{b_n}{x_n} + {c_n}{y_n}} \over {{A_n} + {C_n}{y_n}}}} \hfill \cr } \,\,\,\,\,for} \right.\,\,\,n = 0,1, \ldots , where { c ′ n } n = 0 ∞ \left\{ {{{c'}_n}} \right\}_{n = 0}^\infty , { B ′ n } n = 0 ∞ \left\{ {{{B'}_n}} \right\}_{n = 0}^\infty , { b n } n = 0 ∞ \left\{ {{b_n}} \right\}_{n = 0}^\infty , { c n } n = 0 ∞ \left\{ {{c_n}} \right\}_{n = 0}^\infty , and { A n } n = 0 ∞ \left\{ {{A_n}} \right\}_{n = 0}^\infty are all bounded above and below by positive constants, and { C n } n = 0 ∞ \left\{ {{C_n}} \right\}_{n = 0}^\infty is either bounded above and below by positive constants or is identically zero. In the latter case, we give an example which can be reduced to a system of the form { x n + 1 = x n y n , y n + 1 = x n + γ n y n           f o r       n = 0 , 1 , … , \left\{ {\matrix{ {{x_{n + 1}} = {{{x_n}} \over {{y_n}}},} \hfill \cr {{y_{n + 1}} = {x_n} + {\gamma _n}{y_n}} \hfill \cr } \,\,\,\,\,for} \right.\,\,\,n = 0,1, \ldots , where 0 < γ′ < γ n < γ < 1 for some constants γ and γ′ for all n. This provides a counterexample to the main result of the 2021 paper by Camouzis and Kotsios.

No abstract

No abstract

At a time when funding for programs on academic campuses around the country is tight, financial support for undergraduate research has also become increasingly difficult to find. We discuss some suggestions for funding and supporting undergraduate research programs in mathematics from the 2012 "Trends in Undergraduate Research in Mathematical Sciences" conference held in Chicago,