273,228 research outputs found
Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages
A language over an alphabet is suffix-convex if, for any words
, whenever and are in , then so is .
Suffix-convex languages include three special cases: left-ideal, suffix-closed,
and suffix-free languages. We examine complexity properties of these three
special classes of suffix-convex regular languages. In particular, we study the
quotient/state complexity of boolean operations, product (concatenation), star,
and reversal on these languages, as well as the size of their syntactic
semigroups, and the quotient complexity of their atoms.Comment: 20 pages, 11 figures, 1 table. arXiv admin note: text overlap with
arXiv:1605.0669
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Filling polygonal holes with bicubic patches
Consider a bicubic rectangular patch complex which surrounds an n-sided hole in R3. Then the problem of filling the hole with n bicubic rectangular patches is studied
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The weighted v-spline as a double knot B-spline
The local support basis representation of the ‘weighted v-spline’
is derived in terms of double knot cubic B-splines, so providing a
convenient form for computing and analysing the representation
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Convexity of Bẻzier nets on sub-triangles
This note generalizes a result of Goodman[3], where it is shown that the convexity of Bèzier nets defined on a base triangle is preserved on sub-triangles obtained from a mid-point subdivision process. Here we show that the convexity of Bèzier nets is preserved on and only on sub-triangles that are "parallel" to the base triangle
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Irregular C2 surface construction using bi-polynomial rectangular patches
The construction of C2 surfaces using bi-polynomial
parametric rectangular patches is studied. In particular, the analy-
sis of the C2 continuity conditions for the case of n patches meeting at
an n-vertex is developed
A New Technique for Reachability of States in Concatenation Automata
We present a new technique for demonstrating the reachability of states in
deterministic finite automata representing the concatenation of two languages.
Such demonstrations are a necessary step in establishing the state complexity
of the concatenation of two languages, and thus in establishing the state
complexity of concatenation as an operation. Typically, ad-hoc induction
arguments are used to show particular states are reachable in concatenation
automata. We prove some results that seem to capture the essence of many of
these induction arguments. Using these results, reachability proofs in
concatenation automata can often be done more simply and without using
induction directly.Comment: 23 pages, 1 table. Added missing affiliation/funding informatio
Multi-copy and stochastic transformation of multipartite pure states
Characterizing the transformation and classification of multipartite
entangled states is a basic problem in quantum information. We study the
problem under two most common environments, local operations and classical
communications (LOCC), stochastic LOCC and two more general environments,
multi-copy LOCC (MCLOCC) and multi-copy SLOCC (MCSLOCC). We show that two
transformable multipartite states under LOCC or SLOCC are also transformable
under MCLOCC and MCSLOCC. What's more, these two environments are equivalent in
the sense that two transformable states under MCLOCC are also transformable
under MCSLOCC, and vice versa. Based on these environments we classify the
multipartite pure states into a few inequivalent sets and orbits, between which
we build the partial order to decide their transformation. In particular, we
investigate the structure of SLOCC-equivalent states in terms of tensor rank,
which is known as the generalized Schmidt rank. Given the tensor rank, we show
that GHZ states can be used to generate all states with a smaller or equivalent
tensor rank under SLOCC, and all reduced separable states with a cardinality
smaller or equivalent than the tensor rank under LOCC. Using these concepts, we
extended the concept of "maximally entangled state" in the multi-partite
system.Comment: 8 pages, 1 figure, revised version according to colleagues' comment
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Smooth parametric surfaces and n-sided patches
The theory of 'geometric continuity' within the subject of CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed
Most Complex Non-Returning Regular Languages
A regular language is non-returning if in the minimal deterministic
finite automaton accepting it there are no transitions into the initial state.
Eom, Han and Jir\'askov\'a derived upper bounds on the state complexity of
boolean operations and Kleene star, and proved that these bounds are tight
using two different binary witnesses. They derived upper bounds for
concatenation and reversal using three different ternary witnesses. These five
witnesses use a total of six different transformations. We show that for each
there exists a ternary witness of state complexity that meets the
bound for reversal and that at least three letters are needed to meet this
bound. Moreover, the restrictions of this witness to binary alphabets meet the
bounds for product, star, and boolean operations. We also derive tight upper
bounds on the state complexity of binary operations that take arguments with
different alphabets. We prove that the maximal syntactic semigroup of a
non-returning language has elements and requires at least
generators. We find the maximal state complexities of atoms of
non-returning languages. Finally, we show that there exists a most complex
non-returning language that meets the bounds for all these complexity measures.Comment: 22 pages, 6 figure
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