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DTSTART;TZID=America/Los_Angeles:20220201T123000
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SUMMARY:Niho's last conjecture (Daniel Katz\, Cal State Northridge)
DESCRIPTION:A power permutation of a finite field F is a permutation of F whose functional form is x -> x^d for some exponent d.  Power permutations are used in cryptography\, and the exponent d must be chosen so that the permutation is highly nonlinear\, that is\, not easily approximated by linear functions.  The Walsh spectrum of a power permutation is a list of numbers measuring the correlation of our power permutation with the various linear functions. The last conjecture in Niho’s 1972 thesis considers a particular infinite family of highly nonlinear power permutations\, and states that each permutation in this family has a Walsh spectrum with at most five distinct values. Niho’s own techniques show that there are at most eight distinct values. Each of the eight candidate values corresponds to a possible number of distinct roots of a seventh degree polynomial on a subset of the finite field F called the unit circle. We use symmetry arguments to show that it is impossible to have four\, six\, or seven roots on the unit circle: this proves Niho’s last conjecture. This is joint work with Tor Helleseth and Chunlei Li.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-daniel-katz-cal-state-northridge/
LOCATION:On Zoom
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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DTSTART;TZID=America/Los_Angeles:20220208T123000
DTEND;TZID=America/Los_Angeles:20220208T132000
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CREATED:20220131T003643Z
LAST-MODIFIED:20220131T003643Z
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SUMMARY:Frame coherence and nearly orthogonal lattices (Lenny Fukshansky\, CMC)
DESCRIPTION:A frame in a Euclidean space is a spanning set\, which can be overdetermined. Large frames are used for redundant signal transmission\, which allows for error correction. An important parameter of frames is coherence\, which is maximal absolute value of the cosine of the angle between two frame vectors: the smaller it is\, the closer is the frame to being orthogonal\, which minimizes noise from overlapping frequencies in transmission. One good source frames with sufficiently low coherence comes from layers of minimal vectors in a lattice. We will discuss a particular class of so-called nearly orthogonal lattices\, which exhibits some interesting properties from the stand-point of coherence and other related optimization problems. This is joint work with David Kogan (CGU).
URL:https://colleges.claremont.edu/ccms/event/frame-coherence-and-nearly-orthogonal-lattices-lenny-fukshansky-cmc/
LOCATION:On Zoom
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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DTSTART;TZID=America/Los_Angeles:20220215T123000
DTEND;TZID=America/Los_Angeles:20220215T132000
DTSTAMP:20260411T211456
CREATED:20220119T153839Z
LAST-MODIFIED:20220210T023821Z
UID:2540-1644928200-1644931200@colleges.claremont.edu
SUMMARY:Recent trends in using representations in voting theory - committees and cyclic orders (Karl-Dieter Crisman\, Gordon College)
DESCRIPTION:One of the most important axioms in analyzing voting systems is that of “neutrality”\, which stipulates that the system should treat all candidates symmetrically. Even though this doesn’t always directly apply (such as in primary systems or those with intentional incumbent protection)\, it is extremely important both in theory and practice.If the voting systems in question additionally are tabulated using some sort of points\, we can translate the notion of neutrality into invariance under an action of the symmetric group on a vector space. This means we can exploit representation theory to analyze them\, and this has been successfully done in a number of social choice contexts from cooperative games to voting on full rankings.In this talk\, we describe recent progress in extending this technique to two interesting situations. First we consider work of Barcelo et al. on voting for committees of representatives (such as for departments in a college)\, where the wreath product of two symmetric groups comes into play. Then we look at work by Crisman et al. regarding voting on “cyclic orders”\, or ways to seat people around a table\, which implicitly has both left and right actions of the symmetric group to consider.
URL:https://colleges.claremont.edu/ccms/event/antc-seminar-karl-dieter-crisman-gordon-college/
LOCATION:On Zoom
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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