{"id":514,"date":"2015-05-22T12:42:08","date_gmt":"2015-05-22T12:42:08","guid":{"rendered":"http:\/\/blogs.nd.edu\/parkhillgroup\/?p=514"},"modified":"2015-05-22T12:42:08","modified_gmt":"2015-05-22T12:42:08","slug":"fun-math-problem","status":"publish","type":"post","link":"https:\/\/sites.nd.edu\/parkhillgroup\/2015\/05\/22\/fun-math-problem\/","title":{"rendered":"Fun Math Problem."},"content":{"rendered":"

I just finished a really fun elementary math problem that I needed for my work; thought I’d share the answer. The problem is this: suppose you have 8 positive numbers (p,q,r,s,t,u,v,w), all co-prime besides possibly being equal. How many different types of equality ensure:<\/p>\n

p+q+r+s-t-u-v-w = 0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(1)<\/strong><\/p>\n

For example one solution is (p=t, q=u, r=v, s=w). The answer is related to a\u00a0“rotating-wave”<\/em><\/a> or “secular<\/em>” approximation for a many-particle density matrix in quantum dynamics. We can use sets of numbers which obey this condition to dramatically accelerate our many-body quantum dynamics calculations and treat correlation energy.<\/p>\n

It turns out that the following expression of Kronecker deltas<\/a> is 1 for every p,q,r,s… that satisfies (1) and zero otherwise, as you can easily verify. I’ll leave the derivation as an exercise to the reader :).<\/p>\n

\"secularizer\"<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

I just finished a really fun elementary math problem that I needed for my work; thought I’d share the answer. The problem is this: suppose you have 8 positive numbers (p,q,r,s,t,u,v,w), all co-prime besides possibly being equal. How many different … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1293,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-514","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/posts\/514","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/users\/1293"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/comments?post=514"}],"version-history":[{"count":3,"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/posts\/514\/revisions"}],"predecessor-version":[{"id":518,"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/posts\/514\/revisions\/518"}],"wp:attachment":[{"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/media?parent=514"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/categories?post=514"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.nd.edu\/parkhillgroup\/wp-json\/wp\/v2\/tags?post=514"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}