Interpolation of nonlinear positive or order preserving operators on Banach lattices

Positivity - We study the relationship between exact interpolation spaces for positive, linear operators, for order preserving, Lipschitz continuous operators, and for positive...     

Crystalline Nitridophosphates by Ammonothermal Synthesis

Nitridophosphates are a well-studied class of compounds with high structural diversity. However, their synthesis is quite challenging, particularly due to the limited thermal stability of starting materials like P₃N₅. Typically, it requires even high-pressure techniques (e.g. multianvil) in most cases. Herein, we establish the ammonothermal method as a versatile synthetic tool to access nitridophosphates with different degrees of condensation. α-Li₁₀P₄N₁₀, β-Li₁₀P₄N₁₀, Li₁₈P₆N₁₆, Ca₂PN₃, SrP₈N₁₄, and LiPN₂ were synthesized in supercritical NH₃ at temperatures and pressures up to 1070 K and 200 MPa employing ammonobasic conditions. The products were analyzed by powder X-ray diffraction, energy dispersive X-ray spectroscopy, and FTIR spectroscopy. Moreover, we established red phosphorus as a starting material for nitridophosphate synthesis instead of commonly used and not readily available precursors, such as P₃N₅. This opens a promising preparative access to the emerging compound class of nitridophosphates.     

Scheduling products with subassemblies and changeover time

We revisit the problem, previously studied by Coffman et al, of scheduling products with two subassemblies on a common resource, where changeovers consume time, under the objective of flow-time minimization. We derive some previously unidentified structural properties that could be important to researchers working on similar batch scheduling problems. We show that there exists a series of base schedules from which optimal schedules can be easily derived. As these base schedules build on each other, they are easy to construct as well. We also show that the structure of these base schedules is such that batch sizes decrease over time in a well-defined manner. These insights about the general form of the schedules might also be important to practitioners wanting some intuition about the schedule structure that they are implementing.     

An SR1/BFGS SQP algorithm for nonconvex nonlinear programs with block-diagonal Hessian matrix

We present a quasi-Newton sequential quadratic programming (SQP) algorithm for nonlinear programs in which the Hessian of the Lagrangian function is block-diagonal. Problems with this characteristic frequently arise in the context of optimal control; for example, when a direct multiple shooting parametrization is used. In this article, we describe an implementation of a filter line-search SQP method that computes search directions using an active-set quadratic programming (QP) solver. To take advantage of the block-diagonal structure of the Hessian matrix, each block is approximated separately by quasi-Newton updates. For nonconvex instances, that arise, for example, in optimum experimental design control problems, these blocks are often found to be indefinite. In that case, the block-BFGS quasi-Newton update can lead to poor convergence. The novel aspect in this work is the use of SR1 updates in place of BFGS approximations whenever possible. The resulting indefinite QPs necessitate an inertia control mechanism within the sparse Schur-complement factorization that is carried out by the active-set QP solver. This permits an adaptive selection of the Hessian approximation that guarantees sufficient progress towards a stationary point of the problem. Numerical results demonstrate that the proposed approach reduces the number of SQP iterations and CPU time required for the solution of a set of optimal control problems.     

A Factor Analysis Perspective on Linear Regression in the ‘More Predictors than Samples’ Case

Linear regression (LR) is a core model in supervised machine learning performing a regression task. One can fit this model using either an analytic/closed-form formula or an iterative algorithm. Fitting it via the analytic formula becomes a problem when the number of predictors is greater than the number of samples because the closed-form solution contains a matrix inverse that is not defined when having more predictors than samples. The standard approach to solve this issue is using the Moore–Penrose inverse or the L2 regularization. We propose another solution starting from a machine learning model that, this time, is used in unsupervised learning performing a dimensionality reduction task or just a density estimation one—factor analysis (FA)—with one-dimensional latent space. The density estimation task represents our focus since, in this case, it can fit a Gaussian distribution even if the dimensionality of the data is greater than the number of samples; hence, we obtain this advantage when creating the supervised counterpart of factor analysis, which is linked to linear regression. We also create its semisupervised counterpart and then extend it to be usable with missing data. We prove an equivalence to linear regression and create experiments for each extension of the factor analysis model. The resulting algorithms are either a closed-form solution or an expectation–maximization (EM) algorithm. The latter is linked to information theory by optimizing a function containing a Kullback–Leibler (KL) divergence or the entropy of a random variable.     

Algorithmic Aspects of Tree Amalgamation

The amalgamation of leaf-labeled trees into a single (super)tree that “displays” each of the input trees is an important problem in classification. We discuss various approaches to this problem and show that a simple and well-known polynomial-time algorithm can be used to solve this problem whenever the input set of trees contains a minimum size subset that uniquely determines the supertree. Our results exploit a recently established combinatorial property concerning the structure of such collections of trees.     

Approximation by the Complex form of a Link Operator Between the Phillips and the Szász–Mirakjan Operators

Results in Mathematics - The link operator $${P_{\alpha}^\rho(f,x)=\sum_{k=1}^\infty s_{\alpha,k}(x)\int_0^\infty \theta_{\alpha,k}^\rho(t)f(t)dt+e^{-\alpha x}f(0)}$$ , $${\alpha, \rho >...     

Manifolds with almost nonnegative curvature operator and principal bundles

We study closed manifolds with almost nonnegative curvature operator (ANCO) and derive necessary and/or sufficient conditions for the total spaces of principal bundles over (A)NCO manifolds to admit ANCO connection metrics. In particular, we provide first examples of closed simply connected ANCO manifolds which do not admit metrics with nonnegative curvature operator.     

The NiII,HgII and CuII complexes of 12‐membered‐ring mixed‐donor macrocycles

The structures of di­aqua(1,7‐dioxa‐4‐thia‐10‐aza­cyclo­do­decane)­nickel dinitrate, [Ni(C₈H₁₇NO₂S)(H₂O)₂](NO₃)₂, (I), bis­(nitrato‐O,O′)(1,4,7‐trioxa‐10‐aza­cyclo­do­decane)­mercury, [Hg(NO₃)₂(C₈H₁₇NO₃)], (II), and aqua­(nitrato‐O)(1‐oxa‐4,7,10‐tri­aza­cyclo­do­decane)copper nitrate, [Cu(NO₃)(C₈H₁₉N₃O)(H₂O)]NO₃, (III), reveal each macrocycle binding in a tetradentate manner. The conformations of the ligands in (I) and (III) are the same and distinct from that identified for (II). These differences are in agreement with molecular‐mechanics predictions of ligand conformation as a function of metal‐ion size.