As a perfectionist, I have been struggling to have a perfect PIM (personal information management) solution. For the personal part, I currently use (Exchange-based) Outlook.com. For the professional (institutional) part, things have been so hard. This entry records some of the problems I have encountered so far, and for certain ones, a solution. Expect this entry to be a mess and full of complaints.
Gee Law’s Blog
Sharing data among apps is a common task in modern operating system. Both iOS and Windows Runtime provides central means of transmitting data from one app to another beyond the clipboard. In this entry, I will look at the sharing functionalities in both systems.
Group presentation is a method to express groups as a quotient of a free group and a normal subgroup. In a presentation, there are generators and relators, and the presentation yields a ‘minimalist implementation’ of these generators and relations. Some cases of special interests are finitely generated/related/presented groups. From the definition, finite generation + finite relation is not linguistically equivalent to finite presentation. There has been some discussion on the relation of the three properties for several other algebraic structures. This entry proves that finite generation + finite relation imples finite presentation for groups.
What goes wrong when arrows are reversed in the diagrams for direct product/sum universal properties?
I’m taking an algebra course and regaining my algebra sense (non-sense?). Familiar to us is that direct products [resp. sums] of groups [resp. abelian groups] possess their versions of universal properties. When reviewing these things, I went curious on what would go wrong if the arrows were reversed. Let’s try and see.