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We study the product of two relaxed cutters having a common fixed point. We assume that one of the relaxation parameters is greater than two so that the corresponding relaxed cutter is no longer quasi-nonexpansive, but rather demicontractive. We show that if both of the operators are (weakly/linearly) regular, then under certain conditions, the resulting product inherits the same type of regularity. We then apply these results to proving convergence in the weak, norm and linear sense of algorithms that employ such products.

Silicon photonic integrated circuits (PICs) require cost-effective laser sources that can be monolithically integrated. The low cost and low-temperature solution processability of metal halide perovskites (MHPs) make them attractive alternatives to established III-V compound semiconductors for on-chip laser sources in PICs. Cesium lead bromide (CsPbBr$_3$) perovskites are emerging materials for green light-emitting diodes and lasers. To date, amplified spontaneous emission (ASE) at room temperature has been frequently achieved in CsPbBr$_3$ thin films, while reports on lasing are more limited. Here, we demonstrate a first-order grating distributed feedback (DFB) CsPbBr$_3$ thin-film laser operating at room temperature. Planar hot-pressed (PHP)-CsPbBr$_}$, with a low ASE threshold of 14.5 $\mu$Jcm$^{-2}$ under 0.3 nanosecond (ns) pump pulses, was monolithically integrated into a silicon nitride (Si$_3$N$_4$) waveguide platform via a compatible top-down patterning process. The first-order grating DFB PHP-CsPbBr$_3$ thin-film laser operated at 540 nm in the green spectral region, where III-V lasers have limitations, and exhibited a lasing threshold of 0.755 mJcm$^{-2}$ at room temperature. This work marks a significant step toward utilizing MHPs for on-chip green lasers in PICs for commercial applications., Comment: 21 pages

Designing optoelectronic devices based on cesium lead chloride (CsPbCl${_3}$) perovskites requires accurate values of their optical constants. Unfortunately, experimental data for this material is very limited thus far. Therefore, here, we applied spectroscopic ellipsometry (SE) to measure the complex optical constants of thermally evaporated CsPbCl${_3}$ thin films with different thicknesses on Si/SiO${_2}$ substrates. The data were corroborated with scanning electron microscopy (SEM) images and absorption spectroscopy. An optical dispersion model was developed to derive the complex optical constants and film thicknesses. The Tauc-Lorentz model, in conjunction with two harmonic oscillators, was used to extract the required parameters. The extinction coefficient spectrum exhibited a sharp absorption edge at 411 nm, consistent with the absorption spectrum. In addition, the optical bandgap of the film was calculated from the absorption spectra and SE data. The experimental values agree well with the simulation results, with values of $\sim$ 2.99 eV for different film thicknesses. This work provides fundamental information for designing and modeling CsPbCl3-based optoelectronic devices., Comment: 15 pages

Metal-halide perovskites (MHPs) have exciting optoelectronic properties and are under investigation for various applications, such as photovoltaics, light-emitting diodes, and lasers. An essential step toward exploiting the full potential of this class of materials is their large-scale, on-chip integration with high-resolution, top-down patterning. The development of such patterning methods for perovskite films is challenging because of their ionic behavior and adverse reactions with the solvents used in standard lithography processes. Here, we introduce a versatile and precise method comprising photolithography and reactive ion etching (RIE) processes that can be tuned to accommodate different perovskite compositions and morphologies, including 3D, quasi-2D, and quasi-0D structures. Our method utilizes conventional photoresists at reduced temperatures to create micron-sized features down to 1 $\mu$m, providing high reproducibility from chip to chip. The patterning technique is validated through atomic force microscopy (AFM), X-ray diffraction (XRD), optical spectroscopy, and scanning electron microscopy (SEM). It enables the scalable and high-throughput on-chip monolithic integration of MHPs., Comment: 30 pages

We study in general algebras Gratzer's notion of congruence preserving function, characterizing functions in terms of stability under inverse image of particular Boolean algebras of subsets generated from any subset of the algebra. Weakening Gratzer's notion to only finite index congruences, a similar result holds with lattices of sets. Genereralizing the notion to that of stable preorder preserving function, we extend these characterizations to Boolean algebras and lattices generated from any recognizable subset of the algebra. Our starting point is a result with related flavor on the additive algebra of natural integers which was obtained some years ago. All these results can be visualized in the diagram of Table 1. We finally consider some simple particular conditions on the algebra allowing to get a richer diagram.

We give properties of strict pseudocontractions and demicontractions defined on a Hilbert space, which constitute wide classes of operators that arise in iterative methods for solving fixed point problems. In particular, we give necessary and sufficient conditions under which a convex combination and composition of strict pseudocontractions as well as demicontractions that share a common fixed point is again a strict pseudocontraction or a demicontraction, respectively. Moreover, we introduce a generalized relaxation of composition of demicontraction and give its properties. We apply these properties to prove the weak convergence of a class of algorithms that is wider than the Douglas-Rachford algorithm and projected Landweber algorithms. We have also presented two numerical examples, where we compare the behavior of the presented methods with the Douglas-Rachford method., Comment: 27 pages, 3 figures

We consider a Hilbert space that is a product of a finite number of Hilbert spaces and operators that are represented by "componental operators" acting on the Hilbert spaces that form the product space. We attribute operatorial properties to the componental operators rather than to the full operators. The operatorial properties that we discuss include nonexpansivity, firm non-expansivity, relaxed firm nonexpansivity, averagedness, being a cutter, quasi-nonexpansivity, strong quasi-nonexpansivity, strict quasi-nonexpansivity and contraction. Some relationships between operators whose componental operators have such properties and operators that have these properties on the product space are studied. This enables also to define componental fixed point sets and to study their properties. For componental contractions we offer a variant of the Banach fixed point theorem. Our motivation comes from the desire to extend a fully-simultaneous method that takes into account sparsity of the linear system in order to accelerate convergence [Censor et al., On diagonally relaxed orthogonal projection methods, SIAM J. Sci. Comput. 30 (2008), 473-504]. This was originally applicable to the linear case only and gives rise to an iterative process that uses dfferent componental operators during iterations., Comment: 18 pages. Accepted for publication in: Numerical Functional Analysis and Optimization

In this paper, we analyze the convergence %semi-convergence properties of projected non-stationary block iterative methods (P-BIM) aiming to find a constrained solution to large linear, usually both noisy and ill-conditioned, systems of equations. We split the error of the $k$th iterate into noise error and iteration error, and consider each error separately. The iteration error is treated for a more general algorithm, also suited for solving split feasibility problems in Hilbert space. The results for P-BIM come out as a special case. The algorithmic step involves projecting onto closed convex sets. When these sets are polyhedral, and of finite dimension, it is shown that the algorithm converges linearly. We further derive an upper bound for the noise error of P-BIM. Based on this bound, we suggest a new strategy for choosing relaxation parameters, which assist in speeding up the reconstruction process and improving the quality of obtained images. The relaxation parameters may depend on the noise. The performance of the suggested strategy is shown by examples taken from the field of image reconstruction from projections.

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. An algebra is said to be affine complete if every congruence preserving function is a polynomial function. We show that the algebra of (possibly empty) binary trees whose leaves are labeled by letters of an alphabet containing at least one letter, and the free monoid on an alphabet containing at least two letters are affine complete., Comment: 18 pages

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of binary trees whose leaves are labeled by letters of an alphabet containing at least three letters, a function is congruence preserving if and only if it is polynomial., Comment: 9 pages, 1 figure

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of complete binary trees whose leaves are labeled by letters of an alphabet containing at least three letters a function is congruence preserving if and only if it is polynomial. This exhibits an example of a non commutative and non associative affine complete algebra. As far as we know, it is the first example of such an algebra.

Looking at some monoids and (semi)rings (natural numbers, integers and p-adic integers), and more generally, residually finite algebras (in a strong sense), we prove the equivalence of two ways for a function on such an algebra to behave like the operations of the algebra. The first way is to preserve congruences or stable preorders. The second way is to demand that preimages of recognizable sets belong to the lattice or the Boolean algebra generated by the preimages of recognizable sets by derived unary operation of the algebra (such as translations, quotients,. . . ).

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $S$. We assume that $S=C\cap A^{-1}(Q)$ is the nonempty solution set of a (multiple-set) split convex feasibility problem, where $C$ and $Q$ are both closed and convex subsets of two real Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$, respectively, and the operator $A$ acting between them is linear. We consider a modification of the gradient projection method the main idea of which is to replace at each step the metric projection onto $S$ by another metric projection onto a half-space which contains $S$. We propose three variants of a method for constructing the above-mentioned half-spaces by employing the multiple-set and the split structure of the set $S$. For the split part we make use of the Landweber transform.

Metal-halide perovskites are promising lasing materials for realization of monolithically integrated laser sources, the key components of silicon photonic integrated circuits (PICs). Perovskites can be deposited from solution and require only low temperature processing leading to significant cost reduction and enabling new PIC architectures compared to state-of-the-art lasers realized through costly and inefficient hybrid integration of III-V semiconductors. Until now however, due to the chemical sensitivity of perovskites, no microfabrication process based on optical lithography and therefore on existing semiconductor manufacturing infrastructure has been established. Here, the first methylammonium lead iodide perovskite micro-disc lasers monolithically integrated into silicon nitride PICs by such a top-down process is presented. The lasers show a record low lasing threshold of 4.7 ${\mu}$Jcm$^{-2}$ at room temperature for monolithically integrated lasers, which are CMOS compatible and can be integrated in the back-end-of-line (BEOL) processes.

We consider the split convex feasibility problem in a fixed point setting. Motivated by the well-known CQ-method of Byrne (2002), we define an abstract andweber transform which applies to more general operators than the metric projection. We call the result of this transform a Landweber operator. It turns out that the Landweber transform preserves many interesting properties. For example, the Landweber transform of a (quasi/firmly) nonexpansive mapping is again (quasi/firmly) nonexpansive. Moreover, the Landweber transform of a (weakly/linearly) regular mapping is again (weakly/linearly) regular. The preservation of regularity is important because it leads to (weak/linear) convergence of many CQ-type methods.

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.

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.

Metal-halide perovskites are a class of solution processed materials with remarkable optoelectronic properties such as high photoluminescence quantum yields and long carrier lifetimes, which makes them promising for a wide range of efficient photonic devices. In this work, we demonstrate the first successful integration of a perovskite laser onto a silicon nitride photonic chip. High throughput, low cost optical lithography is used followed by indirect structuring of the perovskite waveguide. We embed methylammonium lead tri-iodide (MAPbI3) in a pre-patterned race-track microresonator and couple the emitted light to an integrated photonic waveguide. We clearly observe the build-up of spectrally narrow lasing modes at room temperature upon a pump threshold fluence of $19.6 \mu Jcm^{-2}$. Our results evidence the possibility of on-chip lasers based on metal-halide perovskites with industry relevance on a commercially available dielectric photonic platform, which is a step forward towards low-cost integrated photonic devices.

A function on an algebra is congruence preserving if, for any congruence, it maps congruent elements to congruent elements. We show that, on a free monoid generated by at least 3 letters, a function from the free monoid into itself is congruence preserving %nonmonogenic if and only if it is of the form $x \mapsto w_0 x w_1 \cdots w_{n-1} x w_n$ for some finite sequence of words $w_0,\ldots,w_n$. We generalize this result to functions of arbitrary arity. This shows that a free monoid with at least three generators is a (noncommutative) affine complete algebra. Up to our knowledge, it is the first (nontrivial) case of a noncommutative affine complete algebra.

Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational polynomials (taking only integral values) and the function giving the least common multiple of $1,2,\ldots,k$. The tool used to obtain these characterizations is "lifting": if $\pi\colon X\to Y$ is a surjective morphism, and $f$ a function on $Y$ a lifting of $f$ is a function $F$ on $X$ such that $\pi\circ F=f\circ\pi$. In this paper we relate the finite and infinite notions by proving that the finite case can be lifted to the infinite one. For $p$-adic and profinite integers we get similar characterizations via lifting. We also prove that lattices of recognizable subsets of $Z$ are stable under inverse image by congruence preserving functions.

We introduce a basis of rational polynomial-like functions $P_0,\ldots,P_{n-1}$ for the free module of functions $Z/nZ\to Z/mZ$. We then characterize the subfamily of congruence preserving functions as the set of linear combinations of the functions $lcm(k)\,P_k$ where $lcm(k)$ is the least common multiple of $2,\ldots,k$ (viewed in $Z/mZ$). As a consequence, when $n\geq m$, the number of such functions is independent of $n$.

Our aim is to present several properties of a Landweber operator and of a Landweber-type operator. These operators are widely used in methods for solving the split feasibility problem and the split common fixed point problem. The presented properties can be used in proofs of convergence of related algorithms.

Projections onto sets are used in a wide variety of methods in optimization theory but not every method that uses projections really belongs to the class of projection methods as we mean it here. Here projection methods are iterative algorithms that use projections onto sets while relying on the general principle that when a family of (usually closed and convex) sets is present then projections (or approximate projections) onto the given individual sets are easier to perform than projections onto other sets (intersections, image sets under some transformation, etc.) that are derived from the given family of individual sets. Projection methods employ projections (or approximate projections) onto convex sets in various ways. They may use different kinds of projections and, sometimes, even use different projections within the same algorithm. They serve to solve a variety of problems which are either of the feasibility or the optimization types. They have different algorithmic structures, of which some are particularly suitable for parallel computing, and they demonstrate nice convergence properties and/or good initial behavior patterns. This class of algorithms has witnessed great progress in recent years and its member algorithms have been applied with success to many scientific, technological, and mathematical problems. This annotated bibliography includes books and review papers on, or related to, projection methods that we know about, use, and like. If you know of books or review papers that should be added to this list please contact us., Comment: Revised version. Accepted for publication in the journal "Optimization"

Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying $f(a)-f(b)\equiv0 \pmod {(a-b)}$ for all $a>b$. We characterize this class of functions via their representations as Newton series. This class, which obviously contains all polynomials with integral coefficients, also contains unexpected functions, for instance all functions $x\mapsto\lfloor e^{1/a}\;a^x\;x!\rfloor$, with $a\in\Z\setminus\{0,1\}$, and a function equal to $\lfloor e\;x!\rfloor$ except on 0. Finally, to study the complement class, we look at functions $\N\to\RR$ which are not uniformly close to any function having integral difference ratios.

We consider lattices of regular sets of non negative integers, i.e. of sets definable in Presbuger arithmetic. We prove that if such a lattice is closed under decrement then it is also closed under many other functions: quotients by an integer, roots, etc.

This paper is concerned with the variational inequality problem (VIP) over the fixed point set of a quasi-nonexpansive operator. We propose, in particular, an algorithm which entails, at each step, projecting onto a suitably chosen half-space, and prove that the sequences it generates converge to the unique solution of the VIP. We also present an application of our result to a hierarchical optimization problem., Comment: Accepted for publication in Numerical Functional Analysis and Optimization

We consider sequential iterative processes for the common fixed point problem of families of cutter operators on a Hilbert space. These are operators that have the property that, for any point x\inH, the hyperplane through Tx whose normal is x-Tx always "cuts" the space into two half-spaces one of which contains the point x while the other contains the (assumed nonempty) fixed point set of T. We define and study generalized relaxations and extrapolation of cutter operators and construct extrapolated cyclic cutter operators. In this framework we investigate the Dos Santos local acceleration method in a unified manner and adopt it to a composition of cutters. For these we conduct convergence analysis of successive iteration algorithms., Comment: Journal of Mathematical Analysis and Applications, accepted for publication

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.

In 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 $m\_1,...,m\_q$) 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 $m\_1,...,m\_q$ as subsequences, i.e. for each $m\_i$, if $m\_i=p\_1,..., p\_k$, the letters $p\_1,..., p\_k$ 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 $O(nq)$., Comment: 12