82,080 research outputs found
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Moduli Spaces and Formal Operads
Let overline{M}_{g,n} be the moduli space of stable algebraic curves of genus
g with n marked points. With the operations which relate the different moduli
spaces identifying marked points, the family (overline{M}_{g,n})_{g,n} is a
modular operad of projective smooth Deligne-Mumford stacks, overline{M}. In
this paper we prove that the modular operad of singular chains
C_*(overline{M};Q) is formal; so it is weakly equivalent to the modular operad
of its homology H_*(overline{M};Q). As a consequence, the "up to homotopy"
algebras of these two operads are the same. To obtain this result we prove a
formality theorem for operads analogous to Deligne-Griffiths-Morgan-Sullivan
formality theorem, the existence of minimal models of modular operads, and a
characterization of formality for operads which shows that formality is
independent of the ground field.Comment: 36 pages (v3: some typographical corrections
Entanglement consumption of instantaneous nonlocal quantum measurements
Relativistic causality has dramatic consequences on the measurability of
nonlocal variables and poses the fundamental question of whether it is
physically meaningful to speak about the value of nonlocal variables at a
particular time. Recent work has shown that by weakening the role of the
measurement in preparing eigenstates of the variable it is in fact possible to
measure all nonlocal observables instantaneously by exploiting entanglement.
However, for these measurement schemes to succeed with certainty an infinite
amount of entanglement must be distributed initially and all this entanglement
is necessarily consumed. In this work we sharpen the characterisation of
instantaneous nonlocal measurements by explicitly devising schemes in which
only a finite amount of the initially distributed entanglement is ever
utilised. This enables us to determine an upper bound to the average
consumption for the most general cases of nonlocal measurements. This includes
the tasks of state verification, where the measurement verifies if the system
is in a given state, and verification measurements of a general set of
eigenstates of an observable. Despite its finiteness the growth of entanglement
consumption is found to display an extremely unfavourable exponential of an
exponential scaling with either the number of qubits needed to contain the
Schmidt rank of the target state or total number of qubits in the system for an
operator measurement. This scaling is seen to be a consequence of the
combination of the generic exponential scaling of unitary decompositions
combined with the highly recursive structure of our scheme required to overcome
the no-signalling constraint of relativistic causality.Comment: 32 pages and 14 figures. Updated to published versio
Visual characterization of associative quasitrivial nondecreasing operations on finite chains
In this paper we provide visual characterization of associative quasitrivial
nondecreasing operations on finite chains. We also provide a characterization
of bisymmetric quasitrivial nondecreasing binary operations on finite chains.
Finally, we estimate the number of functions belonging to the previous classes.Comment: 25 pages, 18 Figure
- …