the example supplied in the question has correct input, except that the author names are usually presented in "normal" order, not surname first (unless, as in chinese, the surname really does come first).
the \thanks associated with each author are placed as unnumbered footnotes at the bottom of the first page. it is usual, with this arrangement, to say something like "The first author was supported by ...", etc., so that it is clear which acknowledgement applies to which author.
however, what i see in the example is probably a url, not a normal acknowledgment. for this use, the ams document classes provide the command \urladdr which, as for e-mail addresses, is listed with the regular address; for amsart, this is at the end of the article.
if footnote markers are required, \footnotemark and \footnotetext can be used judiciously to add the markers manually. however, you may have to manually adjust the value of the counter that is used for the markers to get appropriate results, and you will also need to supply "optional" alternates for the author names to avoid having the footote markers appear in running heads.
the arrangement described is a design decision specific to ams article classes; unless the publication to which you are submitting your article requires amsart, it is possible that a different document class might better match your expectations.
authblkpackage? – Jörg Nov 10 '12 at 09:38amsart, which features a different syntax for\authorand\thanks. – lockstep Nov 10 '12 at 09:41\thanks{Author 1 thanks Xyz}. – egreg Nov 10 '12 at 10:44\thanks] will be printed as an unnumbered footnote at the bottom of the first page of the article." That seems to be happening alright, but no symbolic footnote marks are generated. The lack of footnote markers may be a design decision, as @egreg hypothesizes, or it may be an oversight/bug. – Mico Nov 10 '12 at 11:40\thanksin `amsart (and the other ams document classes) is definitely a design decision. the transition from using a footnote marker to no marker appears to have been made during the period 1986-1988 (before the uniform use of tex to produce ams journals). – barbara beeton Nov 10 '12 at 13:54\footnotemarkand\footnotetextcan be used judiciously to add the markers manually. you may have to manually adjust the value of the counter that is used for the markers to get appropriate results. – barbara beeton Nov 10 '12 at 13:58\thankses is "Network1" and "Network2". if that means what it appears to mean, perhaps the\urladdrprovided byamsartwould be more suitable. however, that, like the addresses, will be placed at the end of the article. it's possible that a different document class might match your expectations better thanamsart, unless that's what is required by the journal to which you are submitting it. – barbara beeton Nov 10 '12 at 14:04