19,319 research outputs found
Paths to Understanding Birational Rowmotion on Products of Two Chains
Birational rowmotion is an action on the space of assignments of rational
functions to the elements of a finite partially-ordered set (poset). It is
lifted from the well-studied rowmotion map on order ideals (equivariantly on
antichains) of a poset , which when iterated on special posets, has
unexpectedly nice properties in terms of periodicity, cyclic sieving, and
homomesy (statistics whose averages over each orbit are constant) [AST11, BW74,
CF95, Pan09, PR13, RuSh12,RuWa15+,SW12, ThWi17, Yil17. In this context,
rowmotion appears to be related to Auslander-Reiten translation on certain
quivers, and birational rowmotion to -systems of type
described in Zamolodchikov periodicity.
We give a formula in terms of families of non-intersecting lattice paths for
iterated actions of the birational rowmotion map on a product of two chains.
This allows us to give a much simpler direct proof of the key fact that the
period of this map on a product of chains of lengths and is
(first proved by D.~Grinberg and the second author), as well as the first proof
of the birational analogue of homomesy along files for such posets.Comment: 31 pages, to appear in Algebraic Combinatoric
The universality of iterated hashing over variable-length strings
Iterated hash functions process strings recursively, one character at a time.
At each iteration, they compute a new hash value from the preceding hash value
and the next character. We prove that iterated hashing can be pairwise
independent, but never 3-wise independent. We show that it can be almost
universal over strings much longer than the number of hash values; we bound the
maximal string length given the collision probability
Relations between elliptic multiple zeta values and a special derivation algebra
We investigate relations between elliptic multiple zeta values and describe a
method to derive the number of indecomposable elements of given weight and
length. Our method is based on representing elliptic multiple zeta values as
iterated integrals over Eisenstein series and exploiting the connection with a
special derivation algebra. Its commutator relations give rise to constraints
on the iterated integrals over Eisenstein series relevant for elliptic multiple
zeta values and thereby allow to count the indecomposable representatives.
Conversely, the above connection suggests apparently new relations in the
derivation algebra. Under https://tools.aei.mpg.de/emzv we provide relations
for elliptic multiple zeta values over a wide range of weights and lengths.Comment: 43 pages, v2:replaced with published versio
Time Quasilattices in Dissipative Dynamical Systems
We establish the existence of `time quasilattices' as stable trajectories in
dissipative dynamical systems. These tilings of the time axis, with two unit
cells of different durations, can be generated as cuts through a periodic
lattice spanned by two orthogonal directions of time. We show that there are
precisely two admissible time quasilattices, which we term the infinite Pell
and Clapeyron words, reached by a generalization of the period-doubling
cascade. Finite Pell and Clapeyron words of increasing length provide
systematic periodic approximations to time quasilattices which can be verified
experimentally. The results apply to all systems featuring the universal
sequence of periodic windows. We provide examples of discrete-time maps, and
periodically-driven continuous-time dynamical systems. We identify quantum
many-body systems in which time quasilattices develop rigidity via the
interaction of many degrees of freedom, thus constituting dissipative discrete
`time quasicrystals'.Comment: 38 pages, 14 figures. This version incorporates "Pell and Clapeyron
Words as Stable Trajectories in Dynamical Systems", arXiv:1707.09333.
Submission to SciPos
A class of non-holomorphic modular forms II : equivariant iterated Eisenstein integrals
We introduce a new family of real analytic modular forms on the upper half
plane. They are arguably the simplest class of `mixed' versions of modular
forms of level one and are constructed out of real and imaginary parts of
iterated integrals of holomorphic Eisenstein series. They form an algebra of
functions satisfying many properties analogous to classical holomorphic modular
forms. In particular, they admit expansions in and
involving only rational numbers and single-valued multiple zeta values. The
first non-trivial functions in this class are real analytic Eisenstein series.Comment: Introduction rewritten in version 2, and other minor edit
Iterative Universal Rigidity
A bar framework determined by a finite graph and configuration in
space is universally rigid if it is rigid in any . We provide a characterization of universally rigidity for any
graph and any configuration in terms of a sequence of affine
subsets of the space of configurations. This corresponds to a facial reduction
process for closed finite dimensional convex cones.Comment: 41 pages, 12 figure
Spectrum of Fractal Interpolation Functions
In this paper we compute the Fourier spectrum of the Fractal Interpolation
Functions FIFs as introduced by Michael Barnsley. We show that there is an
analytical way to compute them. In this paper we attempt to solve the inverse
problem of FIF by using the spectru
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