The Bloom filter is a space efficient randomized data structure for representing a set and supporting membership queries. Bloom filters intrinsically allow false positives. However, the space savings they offer outweigh the disadvantage if the false positive rates are kept sufficiently low. Inspired by the recent application of the Bloom filter in a novel multicast forwarding fabric, this paper proposes a variant of the Bloom filter, the optihash. The optihash introduces an optimization for the false positive rate at the stage of Bloom filter formation using the same amount of space at the cost of slightly more processing than the classic Bloom filter. Often Bloom filters are used in situations where a fixed amount of space is a primary constraint. We present the optihash as a good alternative to Bloom filters since the amount of space is the same and the improvements in false positives can justify the additional processing. Specifically, we show via simulations and numerical analysis that using the optihash the false positives occurrences can be reduced and controlled at a cost of small additional processing. The simulations are carried out for in-packet forwarding. In this framework, the Bloom filter is used as a compact link/route identifier and it is placed in the packet header to encode
Mathematical mindset theory suggests learner motivation in mathematics may be increased by opening problems using a set of recommended ideas. However, very little evidence supports this theory. We explore motivation through self-reports while learners attempt problems formulated according to mindset theory and standard problems. We also explore neural correlates of motivation and felt-affect while participants attempt the problems. Notably, we do not tell participants what mindset theory is and instead simply investigate whether mindset problems affect reported motivation levels and neural correlates of motivation in learners. We find significant increases in motivation for mindset problems compared to standard problems. We also find significant differences in brain activity in prefrontal EEG asymmetry between problems. This provides some of the first evidence that mathematical mindset theory increases motivation (even when participants are not aware of mindset theory), and that this change is reflected in brain activity of learners attempting mathematical problems.
Abstract. We introduce a knot semigroup as a cancellative semigroup whose dening relations are produced from crossings on a knot diagram in a way similar to the Wirtinger presentation of the knot group; to be more precise, a knot semigroup as we dene it is closely related to such tools of knot theory as the 2-fold branched cyclic cover space of a knot and the involutory quandle of a knot. We describe knot semigroups of several standard classes of knot diagrams, including torus knots and torus links T (2, n) and twist knots. The description includes a solution of the word problem. To produce this description, we introduce alternating sum semigroups as certain naturally dened factor semigroups of free semigroups over cyclic groups. We formulate several conjectures for future research. The context and the paper planWe consider cancellative semigroups (which we call knot semigroups) whose dening relations come in pairs of the form xy = yz and yx = zy, where x, y, z are generators, and are`read' from a certain natural diagram (namely, a knot diagram 1 ). Our inspiration in this research comes partially from the study of right-angled Artin groups (and the corresponding semigroup-theory construction, trace monoids [1]). A right-angled Artin group is a group in which every dening relation has a form xy = yx. Given an undirected graph, one can dene a group whose set of generators is the set of vertices of the graph, and a dening relation xy = yx is introduced whenever vertices x and y are adjacent. This construction denes a natural correspondence between undirected graphs and right-angled Artin groups. Whereas knot diagrams are not as ubiquitous as graphs, they have attracted much attention of algebraists in the last century, and knot semigroups described in this paper can become a new natural way of dening semigroups corresponding to knot diagrams; we discuss this further in Section 8. Each relation dening a knot semigroup has words of the same length on the two sides of the equality; such relations are called homogeneous and, accordingly, semigroups dened in this way are also sometimes called homogeneous; another example of homogeneous semigroups are braid semigroups (for their denition see, for example, [2]); for a brief review of more examples of classes of homogeneous semigroups see [3]. There has been a number of attempts to dene conjugate elements in semigroups, generalising conjugation in groups. For recent reviews of ideas in this direction, see 1 Note that a knot semigroup is not a knot invariant; that is, there are cases when two dierent diagrams of the same knot produce non-isomorphic knot semigroups. See more in Section 8.
Given an integer $n$, we show that $\mathcal{I}_{n}$ embeds in a 2-generated subsemigroup of $\mathcal{I}_{n+2}$, which is an inverse semigroup. An immediate consequence of this result is the following, which is analogous to the case for groups and semigroups: every finite inverse semigroup may be embedded in a finite 2-generated semigroup which is an inverse semigroup.AMS 2000 Mathematics subject classification: Primary 20M18. Secondary 20M20
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