tag:blogger.com,1999:blog-5967982571807204356.post6646535461264018570..comments2015-12-17T20:53:47.229-08:00Comments on Mathematics in Research: Functionsekaveerahttp://www.blogger.com/profile/12705417398875109678noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-5967982571807204356.post-13410160193572418822015-12-17T20:53:47.229-08:002015-12-17T20:53:47.229-08:00The $L(x)$ in my earlier comment should be $L(x) =...The $L(x)$ in my earlier comment should be $L(x) = ... m(4x) ... $ that is, it is infinite in both directionsfrogeyedpeashttps://www.blogger.com/profile/14765386764816634435noreply@blogger.comtag:blogger.com,1999:blog-5967982571807204356.post-60296954619665904292015-12-17T20:53:03.109-08:002015-12-17T20:53:03.109-08:00We can engineer a solution pretty easily here. Con...We can engineer a solution pretty easily here. Consider some $m(x)$ (we'll talk about what this is later).<br /><br />Consider <br />$$L(x)= m(4x) + m(2x) + m(x) + m(\frac{x}{2}) + m(\frac{x}{8}) + ... $$ <br /><br />It's easy to see that formally speaking $$L(x/2) = L(x)$$. Now that series I gave seems hopeless at first since it consider values $m$ at 0 and infinity. That is fine, let us consider $m(x) = x^3e^{-x^2}$. This rapidly goes to 0 as the $|x|$ gets very large, as well as very small. Then substituting that back into the definition of $L$ gives you a function that is periodic with respect to doubling the period (at least over the reals). frogeyedpeashttps://www.blogger.com/profile/14765386764816634435noreply@blogger.comtag:blogger.com,1999:blog-5967982571807204356.post-6706469337350403382014-04-18T06:11:06.161-07:002014-04-18T06:11:06.161-07:00In general $$f(x)=Const$$ is one such function. In general $$f(x)=Const$$ is one such function. Anonymoushttps://www.blogger.com/profile/17454655166620369091noreply@blogger.comtag:blogger.com,1999:blog-5967982571807204356.post-66208344262810646432014-04-16T00:34:02.963-07:002014-04-16T00:34:02.963-07:00If it is true for all x, then I think it would be ...If it is true for all x, then I think it would be f(x)=0.<br />Sawarnikhttps://www.blogger.com/profile/11624894741269891041noreply@blogger.com