Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages

A language $L$ over an alphabet $\Sigma$ is suffix-convex if, for any words $x,y,z\in\Sigma^*$, whenever $z$ and $xyz$ are in $L$, then so is $yz$. 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

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

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

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

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

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 $L$ 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 $n\ge 4$ there exists a ternary witness of state complexity $n$ 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 $(n-1)^n$ elements and requires at least $\binom{n}{2}$ 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