for all . Thus for instance and are (trivially) upper bound sieves and lower bound sieves respectively. (i) The supremal value of the quantity (1), subject to the constraints in Problem 3, is equal to the infimal value of the quantity , as ranges over all upper bound sieves. (ii...
Slightly more non-trivially, the sum set estimate established by Ruzsa, has an entropy analogue , and similarly for a number of other standard sumset inequalities in the literature (e.g. the Rusza triangle inequality, the Plünnecke-Rusza inequality, and the Balog-Szemeredi-Gowers theorem, ...
Starting early with sharing documents with a teacher is an excellent practice. Google makes this trivially easy. We can see what students are producing at the various stages of its development. This avoids the shock of the final piece arriving fully formed on the deadline day. It affords oppor...
I hope you realize that I could do so trivially by defining the term "significance" any way I see fit, exploiting yet another fallacy in your reasoning? Aug 4, 2014 #38 A.T. Science Advisor 12,571 3,715 voko said: ...defining the term "significance" any way I see fit......
etc. MapReduce provides thegroup byoperationdirectly (in the shuffle and reduce phases), and it provides theorder byoperationindirectly through the way it implements the grouping. Filtering and projection can be implemented trivially in the map phase. But other operators—particularlyjoin—are not pr...
Presumably there is a sense in which two people could carry out this exercise, with only one of them “getting it right”, and the other “getting it wrong”. My question is what does “getting it right” mean to you? I’m not asking about particular inferential outputs, like point/int...
. These are called the “trivial zeros” because it’s trivially easy* to prove that they’re there. There are also other zeros called the “non-trivial zeros” that extend on, or very near, a line perpendicular to the real numbers. In fact, the statement of the Riemann hypothesis...
On the left-hand side is the product of the expansion of each exponential while on the right-hand side it is the expansion of the single exponential (no-trivially) combined taking into account the commutator of the exponents on the left-hand side. Verify that both expansions are one and th...
John Schuhr, while this approach does work, it is ultimately no different than rolling a d8 and declaring that the 8 is a reroll. However, this does open up a different mathematical possibility if you construct Turing machines. Since a d2 could be converted to binary, and compute...
He finds that one definition in which determinism is conflated with predictability renders determinism trivially false. Considering the other definition that avoids the conflation, Bricmont asks a pertinent question: Is there a function—in a Platonic sense (that is, independent of our ignorance)—...