1,101,081 research outputs found
Faster Algorithms for Computing Plurality Points
Let V be a set of n points in R^d, which we call voters, where d is a fixed constant. A point p in R^d is preferred over another point p\u27 in R^d by a voter v in V if dist(v,p) < dist(v,p\u27). A point p is called a plurality point if it is preferred by at least as many voters as any other point p\u27.
We present an algorithm that decides in O(n log n) time whether V admits a plurality point in the L_2 norm and, if so, finds the (unique) plurality point. We also give efficient algorithms to compute the smallest subset W of V such that V - W admits a plurality point, and to compute a so-called minimum-radius plurality ball.
Finally, we consider the problem in the personalized L_1 norm, where each point v in V has a preference vector and the distance from v to any point p in R^d is given by sum_{i=1}^d w_i(v) cdot |x_i(v)-x_i(p)|. For this case we can compute in O(n^(d-1)) time the set of all plurality points of V. When all preference vectors are equal, the running time improves to O(n)
The endomorphism ring of the trivial module in a localized category
Suppose that is a finite group and is a field of characteristic . Let be the thick tensor ideal of finitely generated modules
whose support variety is in a fixed subvariety of the projectivized prime
ideal spectrum . Let
denote the Verdier localization of the stable module category
at . We show that if is a finite
collection of closed points and if the -rank every maximal elementary
abelian -subgroups of is at least 3, then the endomorphism ring of the
trivial module in is a local ring whose unique maximal ideal is
infinitely generated and nilpotent. In addition, we show an example where the
endomorphism ring in of a compact object is not finitely
presented as a module over the endomorphism ring of the trivial module.Comment: 21 page
A Combinatorial classification of postcritically fixed Newton maps
We give a combinatorial classification for the class of postcritically fixed
Newton maps of polynomials as dynamical systems. This lays the foundation for
classification results of more general classes of Newton maps.
A fundamental ingredient is the proof that for every Newton map
(postcritically finite or not) every connected component of the basin of an
attracting fixed point can be connected to through a finite chain of
such components.Comment: 37 pages, 5 figures, published in Ergodic Theory and Dynamical
Systems (2018). This is the final author file before publication. Text
overlap with earlier arxiv file observed by arxiv system relates to an
earlier version that was erroneously uploaded separately. arXiv admin note:
text overlap with arXiv:math/070117
Symplectic polarities of buildings of type E₆
A symplectic polarity of a building Delta of type E (6) is a polarity whose fixed point structure is a building of type F (4) containing residues isomorphic to symplectic polar spaces. In this paper, we present two characterizations of such polarities among all dualities. Firstly, we prove that, if a duality theta of Delta never maps a point to a neighbouring symp, and maps some element to a non-opposite element, then theta is a symplectic duality. Secondly, we show that, if a duality theta never maps a chamber to an opposite chamber, then it is a symplectic polarity. The latter completes the programme for dualities of buildings of type E (6) of determining all domestic automorphisms of spherical buildings, and it also shows that symplectic polarities are the only polarities in buildings of type E (6) for which the Phan geometry is empty
The maximum number of systoles for genus two Riemann surfaces with abelian differentials
In this article, we provide bounds on systoles associated to a holomorphic
-form on a Riemann surface . In particular, we show that if
has genus two, then, up to homotopy, there are at most systolic loops on
and, moreover, that this bound is realized by a unique translation
surface up to homothety. For general genus and a holomorphic 1-form
with one zero, we provide the optimal upper bound, , on the
number of homotopy classes of systoles. If, in addition, is hyperelliptic,
then we prove that the optimal upper bound is .Comment: 41 page
Fixed point theorems for Boolean networks expressed in terms of forbidden subnetworks
We are interested in fixed points in Boolean networks, {\em i.e.} functions
from to itself. We define the subnetworks of as the
restrictions of to the subcubes of , and we characterizes a
class of Boolean networks satisfying the following property:
Every subnetwork of has a unique fixed point if and only if has no
subnetwork in . This characterization generalizes the fixed point
theorem of Shih and Dong, which asserts that if for every in
there is no directed cycle in the directed graph whose the adjacency matrix is
the discrete Jacobian matrix of evaluated at point , then has a
unique fixed point. Then, denoting by (resp. )
the networks whose the interaction graph is a positive (resp. negative) cycle,
we show that the non-expansive networks of are exactly the
networks of ; and for the class of
non-expansive networks we get a "dichotomization" of the previous forbidden
subnetwork theorem: Every subnetwork of has at most (resp. at least) one
fixed point if and only if has no subnetworks in (resp.
) subnetwork. Finally, we prove that if is a conjunctive
network then every subnetwork of has at most one fixed point if and only if
has no subnetwork in .Comment: 40 page
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