Regular Sequences of Quasi-Nonexpansive Operators and Their Applications

In this paper we present a systematic study of regular sequences of quasi-nonexpansive operators in Hilbert space. We are interested, in particular, in weakly, boundedly and linearly regular sequences of operators. We show that the type of the regularity is preserved under relaxations, convex combinations and products of operators. Moreover, in this connection, we show that weak, bounded and linear regularity lead to weak, strong and linear convergence, respectively, of various iterative methods. This applies, in particular, to block iterative and string averaging projection methods, which, in principle, are based on the above-mentioned algebraic operations applied to projections. Finally, we show an application of regular sequences of operators to variational inequality problems

On the additive theory of prime numbers II

The undecidability of the additive theory of primes (with identity) as well as the theory Th(N,+, n -> p\_n), where p\_n denotes the (n+1)-th prime, are open questions. As a possible approach, we extend the latter theory by adding some extra function. In this direction we show the undecidability of the existential part of the theory Th(N, +, n -> p\_n, n -> r\_n), where r\_n is the remainder of p\_n divided by n in the euclidian division

Tree inclusions in windows and slices

$P$ is an embedded subtree of $T$ if $P$ can be obtained by deleting some nodes from $T$: if a node $v$ is deleted, all edges adjacent to $v$ are also deleted, and outgoing edges are replaced by edges going from the parent of $v$ (if it exists) to the children of $v$. Deciding whether $P$ is an embedded subtree of $T$ is known to be NP-complete. Given two trees (a target $T$ and a pattern $P$) and a natural number $w$, we address two problems: 1. counting the number of windows of $T$ having height exactly $w$ and containing pattern $P$ as an embedded subtree, and 2. counting the number of slices of $T$ having height exactly $w$ and containing pattern $P$ as an embedded subtree

Overlay Accuracy Limitations of Soft Stamp UV Nanoimprint Lithography and Circumvention Strategies for Device Applications

In this work multilevel pattering capabilities of Substrate Conformal Imprint Lithography (SCIL) have been explored. A mix & match approach combining the high throughput of nanoimprint lithography with the excellent overlay accuracy of electron beam lithography (EBL) has been exploited to fabricate nanoscale devices. An EBL system has also been utilized as a benchmarking tool to measure both stamp distortions and alignment precision of this mix & match approach. By aligning the EBL system to 20 mm x 20 mm and 8 mm x 8 mm cells to compensate pattern distortions of order of $3 \mu m$ over 6 inch wafer area, overlay accuracy better than $1.2 \mu m$ has been demonstrated. This result can partially be attributed to the flexible SCIL stamp which compensates deformations caused by the presence of particles which would otherwise significantly reduce the alignment precision

Multiple serial episode matching

12In a previous paper we generalized the Knuth-Morris-Pratt (KMP) pattern matching algorithm and defined a non-conventional kind of RAM, the MP--RAMs (RAMS equipped with extra operations), and designed an $O(n)$ on-line algorithm for solving the serial episode matching problem on MP--RAMs when there is only one single episode. We here give two extensions of this algorithm to the case when we search for several patterns simultaneously and compare them. More preciseley, given $q+1$ strings (a text $t$ of length $n$ and $q$ patterns $m_1,\ldots,m_q$) and a natural number $w$, the {\em multiple serial episode matching problem} consists in finding the number of size $w$ windows of text $t$ which contain patterns $m_1,\ldots,m_q$ as subsequences, i.e. for each $m_i$, if $m_i=p_1,\ldots ,p_k$, the letters $p_1,\ldots ,p_k$ occur in the window, in the same order as in $m_i$, but not necessarily consecutively (they may be interleaved with other letters).} The main contribution is an algorithm solving this problem on-line in time $O(nq)$

Linear Superiorization for Infeasible Linear Programming

Linear superiorization (abbreviated: LinSup) considers linear programming (LP) problems wherein the constraints as well as the objective function are linear. It allows to steer the iterates of a feasibility-seeking iterative process toward feasible points that have lower (not necessarily minimal) values of the objective function than points that would have been reached by the same feasiblity-seeking iterative process without superiorization. Using a feasibility-seeking iterative process that converges even if the linear feasible set is empty, LinSup generates an iterative sequence that converges to a point that minimizes a proximity function which measures the linear constraints violation. In addition, due to LinSup's repeated objective function reduction steps such a point will most probably have a reduced objective function value. We present an exploratory experimental result that illustrates the behavior of LinSup on an infeasible LP problem.Comment: arXiv admin note: substantial text overlap with arXiv:1612.0653