44,373 research outputs found
Gathering an even number of robots in an odd ring without global multiplicity detection
We propose a gathering protocol for an even number of robots in a ring-shaped
network that allows symmetric but not periodic configurations as initial
configurations, yet uses only local weak multiplicity detection. Robots are
assumed to be anonymous and oblivious, and the execution model is the non-
atomic CORDA model with asynchronous fair scheduling. In our scheme, the number
of robots k must be greater than 8, the number of nodes n on a network must be
odd and greater than k+3. The running time of our protocol is O(n2)
asynchronous rounds.Comment: arXiv admin note: text overlap with arXiv:1104.566
Interview with Kenneth Arrow
Arrow argues that the biggest failures of economic theory are: our failure to explain the business cycle; the missing explanations for the size of fluctuations of prices; our failure to explain the causes of growth and of the spread of innovation. He then discusses several of the existing alternatives to the rational expectations paradigm. He tells the story of his dissertation, and how Koopmans wanted to decline his Nobel Prize.Finally, he discusses health care reform, and malaria in Africa.Health Care; Business Cycles; Fluctuations
A special sequence and primorial numbers
In this paper, we study a class of functions defined recursively on the set
of natural numbers in terms of the greatest common divisor algorithm of two
numbers and requiring a minimality condition. These functions are permutations,
products of infinitely many cycles that depend on certain breaks in the natural
numbers that involve the primes and some special products of primes that have a
density of approximately 29.4%. Knowing more about the main class of these
functions may bring efficient ways in detecting the primality of a given
positive integer.Comment: 12 pages and 2 figure
Embedded in These Walls
Embedded In These Walls uses photographic imagery, archival ephemera, and written text to examine a specific history of generational trauma through the lens of a singular family of a southern tradition to point to a larger systemic breakdown of accountability and truthfulness regarding abus
On Monotone Sequences of Directed Flips, Triangulations of Polyhedra, and Structural Properties of a Directed Flip Graph
This paper studied the geometric and combinatorial aspects of the classical
Lawson's flip algorithm in 1972. Let A be a finite set of points in R2, omega
be a height function which lifts the vertices of A into R3. Every flip in
triangulations of A can be associated with a direction. We first established a
relatively obvious relation between monotone sequences of directed flips
between triangulations of A and triangulations of the lifted point set of A in
R3. We then studied the structural properties of a directed flip graph (a
poset) on the set of all triangulations of A. We proved several general
properties of this poset which clearly explain when Lawson's algorithm works
and why it may fail in general. We further characterised the triangulations
which cause failure of Lawson's algorithm, and showed that they must contain
redundant interior vertices which are not removable by directed flips. A
special case if this result in 3d has been shown by B.Joe in 1989. As an
application, we described a simple algorithm to triangulate a special class of
3d non-convex polyhedra. We proved sufficient conditions for the termination of
this algorithm and show that it runs in O(n3) time.Comment: 40 pages, 35 figure
Holes in I^n
Let F be an arbitrary field of characteristic not 2. We write W(F) for the
Witt ring of F, consisting of the isomorphism classes of all anisotropic
quadratic forms over F. For any element x of W(F), dimension dim x is defined
as the dimension of a quadratic form representing x. The elements of all even
dimensions form an ideal denoted I(F). The filtration of the ring W(F) by the
powers I(F)^n of this ideal plays a fundamental role in the algebraic theory of
quadratic forms. The Milnor conjectures, recently proved by Voevodsky and
Orlov-Vishik-Voevodsky, describe the successive quotients I(F)^n/I(F)^{n+1} of
this filtration, identifying them with Galois cohomology groups and with the
Milnor K-groups modulo 2 of the field F. In the present article we give a
complete answer to a different old-standing question concerning I(F)^n, asking
about the possible values of dim x for x in I(F)^n. More precisely, for any
positive integer n, we prove that the set dim I^n of all dim x for all x in
I(F)^n and all F consisists of 2^{n+1}-2^i, i=1,2,...,n+1 together with all
even integers greater or equal to 2^{n+1}. Previously available partial
informations on dim I^n include the classical Arason-Pfister theorem, saying
that no integer between 0 and 2^n lies in dim I^n, as well as a recent Vishik's
theorem, saying the same on the integers between 2^n and 2^n+2^{n-1} (the case
n=3 is due to Pfister, n=4 to Hoffmann). Our proof is based on computations in
Chow groups of powers of projective quadrics (involving the Steenrod
operations); the method developed can be also applied to other types of
algebraic varieties.Comment: 29 page
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