59 research outputs found
THE BISECTION METHOD: A SINGLE TECHNIQUE YIELDING SIMPLE PROOFS OF THE FOUR HARD THEOREMS OF CALCULUS
The Intermediate Value Theorem, Boundedness Theorem, Extreme Value Theorem, and Integrability Theorem are all vital theorems of Calculus, yet are almost never proved in calculus textbooks. I will give simple proofs of these theorems using the Bisection Method, explain how I incorporated these into inquiry-style lessons at IMSA, and discuss the benefits to students. As a connection to Computer Science, the Bisection Method can be viewed as a Binary Search Algorithm
Tinkertoys for the Twisted D-Series
We study 4D N=2 superconformal field theories that arise from the
compactification of 6D N=(2,0) theories of type D_N on a Riemann surface, in
the presence of punctures twisted by a Z_2 outer automorphism. Unlike the
untwisted case, the family of SCFTs is in general parametrized, not by M_{g,n},
but by a branched cover thereof. The classification of these SCFTs is carried
out explicitly in the case of the D_4 theory, in terms of three-punctured
spheres and cylinders, and we provide tables of properties of twisted punctures
for the D_5 and D_6 theories. We find realizations of Spin(8) and Spin(7) gauge
theories with matter in all combinations of vector and spinor representations
with vanishing beta-function, as well as Sp(3) gauge theories with matter in
the 3-index traceless antisymmetric representation.Comment: 75 pages, 270 figure
Tinkertoys for the Twisted Theory
We study superconformal field theories that arise as the
compactification of the six-dimensional theory of type on a
punctured Riemann surface in the presence of outer-automorphism
twists. We explicitly carry out the classification of these theories in terms
of three-punctured spheres and cylinders, and provide tables of properties of
the -twisted punctures. An expression is given for the
superconformal index of a fixture with twisted punctures of type , which
we use to check our identifications. Several of our fixtures have Higgs
branches which are isomorphic to instanton moduli spaces, and we find that
S-dualities involving these fixtures imply interesting isomorphisms between
hyperK\"ahler quotients of these spaces. Additionally, we find families of
fixtures for which the Sommers-Achar group, which was previously a Coulomb
branch concept, acts non-trivially on the Higgs branch operators.Comment: 52 pages, 56 figure
Seiberg-Witten for with Spinors
supersymmetric gauge theory admits hypermultiplets
in spinor representations of the gauge group, compatible with , for
. The theories with can be obtained as mass-deformations of
the theories, so it is of greatest interest to construct the
theories. In previous works, we discussed the theories.
Here, we turn to the cases. By compactifying the (2,0)
theory on a 4-punctured sphere, we find Seiberg-Witten solutions to almost all
of the remaining cases. There are five theories, however, which do not seem to
admit a realization from six dimensions.Comment: 28 pages, 54 figure
MASTERY BASED GRADING FOR SECONDARY MATHEMATICS
Dr. Trimm will discuss in detail his design and implementation of a mastery grading system in calculus at IMSA and how it offers many benefits over traditional grading. Dr. Trimm will also explain how it makes creating assessments and grading easier and less work for the teacher, while being more accurate
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Twisted and exceptional Tinkertoys for Gaiotto duality
textA large class of 4d N = 2 superconformal field theories arise as compactifications of a 6d (2, 0) theory of type j = A, D, E on a punctured Riemann surface, C. These theories can be classified by listing the allowed fixtures and cylinders which can occur in a pants decomposition of C, and giving the rules for gluing them together. Different pants decompositions of the same surface give different weakly-coupled presentations of the same underlying SCFT, related by S-duality. An even larger class of theories can be constructed in this way by including "twisted" punctures, which carry a non-trivial action of the outer-automorphism group of j. In this dissertation, we discuss the classification procedure for twisted theories of type D [subscript N] , as well as for twisted and untwisted theories of type E₆. Using these results, we write the Seiberg-Witten solutions for all Spin(n) gauge theories with matter in spinor representations which can be realized by compactifying the (2, 0) theory. We also study a family of SCFTs arising from the twisted A [subscript 2N] series, whose twisted punctures are still not fully-understood.Physic
Tinkertoys for the E7 Theory
We classify the class theories of type . These are four-dimensional
superconformal field theories arising from the compactification
of the theory on a punctured Riemann surface, . The
classification is given by listing all 3-punctured spheres ("fixtures"), and
connecting cylinders, which can arise in a pants-decomposition of . We find
exactly 11,000 fixtures with three regular punctures, and an additional 48 with
one "irregular puncture" (in the sense used in our previous works). To organize
this large number of theories, we have created a web application at
https://golem.ph.utexas.edu/class-S/E7/ . Among these theories, we find 10 new
ones with a simple exceptional global symmetry group, as well as a new rank-2
SCFT and several new rank-3 SCFTs. As an application, we study the
strong-coupling limit of the gauge theory with 3 hypermultiplets in the
. Using our results, we also verify recent conjectures that the
compactification of certain theories can alternatively be realized
in class as fixtures in the or theories.Comment: Fixed one entry in table of interacting fixtures with an irregular
punctur
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