On a borderline between the NP-hard and polynomial-time solvable cases of the flow shop with job-dependent storage requirements

The paper is concerned with the two-machine flow shop, where each job requires an additional resource (referred to as storage space) from the start of its first operation till the end of its second operation. The storage requirement of a job is determined by the processing time of its first operation. At any point in time, the total consumption of this additional resource cannot exceed a given limit (referred to as the storage capacity). The goal is to minimise the makespan, i.e. to minimise the time needed for the completion of all jobs. This problem is NP-hard in the strong sense. The paper analyses how the parameter - a lower bound on the storage capacity specified in terms of the processing times, affects the computational complexity.

     

Bounds on the index of rotationally symmetric self-shrinking tori

Geometriae Dedicata - A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by...     

The Cauchy Relations in Linear Elasticity Theory

In linear elasticity, we decompose the elasticity tensor into two irreducible pieces with 15 and 6 independent components, respectively. The vanishing of the piece with 6 independent components corresponds to the Cauchy relations. Thus, for the first time, we recognize the group-theoretical underpinning of the Cauchy relations.     

Annulation of pyrrole ring in 4-acylpyridine-3,5-dicarbonitriles in the presence of ammonia

Chemistry of Heterocyclic Compounds - 4-Acyl-2-amino-6-chloropyridine-3,5-dicarbonitriles underwent heterocyclization in the presence of ammonia in aqueous dioxane medium, involving the...     

A feasibility study of X‐ray phase‐contrast mammographic tomography at the Imaging and Medical beamline of the Australian Synchrotron

Results are presented of a recent experiment at the Imaging and Medical beamline of the Australian Synchrotron intended to contribute to the implementation of low‐dose high‐sensitivity three‐dimensional mammographic phase‐contrast imaging, initially at synchrotrons and subsequently in hospitals and medical imaging clinics. The effect of such imaging parameters as X‐ray energy, source size, detector resolution, sample‐to‐detector distance, scanning and data processing strategies in the case of propagation‐based phase‐contrast computed tomography (CT) have been tested, quantified, evaluated and optimized using a plastic phantom simulating relevant breast‐tissue characteristics. Analysis of the data collected using a Hamamatsu CMOS Flat Panel Sensor, with a pixel size of 100 µm, revealed the presence of propagation‐based phase contrast and demonstrated significant improvement of the quality of phase‐contrast CT imaging compared with conventional (absorption‐based) CT, at medically acceptable radiation doses.     

Cyanide Complexes Based on {Mo6I8}4+ and {W6I8}4+ Cluster Cores

Compounds based on new cyanide cluster anions [{Mo₆I₈}(CN)₆]^2–, trans-[{Mo₆I₈}(CN)₄(MeO)₂]^2– and trans-[{W₆I₈}(CN)₂(MeO)₄]²⁻ were synthesized using mechanochemical or solvothermal synthesis. The crystal and electronic structures as well as spectroscopic properties of the anions were investigated. It was found that the new compounds exhibit red luminescence upon excitation by UV light in the solid state and solutions, as other cluster complexes based on {Mo₆I₈}⁴⁺ and {W₆I₈}⁴⁺ cores do. The compounds can be recrystallized from aqueous methanol solutions; besides this, it was shown using NMR and UV-Vis spectroscopy that anions did not undergo hydrolysis in the solutions for a long time. These facts indicate that hydrolytic stabilization of {Mo₆I₈} and {W₆I₈} cluster cores can be achieved by coordination of cyanide ligands.     

Beliefs and engagement structures: behind the affective dimension of mathematical learning

Beliefs influencing students’ mathematical learning and problem solving are structured and intertwined with larger affective and cognitive structures. This theoretical article explores a psychological concept we term an engagement structure, with which beliefs are intertwined. Engagement structures are idealized, hypothetical constructs, analogous in many ways to cognitive structures. They describe complex “in the moment” affective and social interactions as students work on conceptually challenging mathematics. We present engagement structures in a self-contained way, paying special attention to their theoretical justification and relation to other constructs. We suggest how beliefs are characteristically woven into their fabric and influence their activation. The research is based on continuing studies of middle school students in inner-city classrooms in the USA.     

Monocyclic and Fused 4-Imino(4-Oxo)-1,3,2-Diazaphospholanes and - Phosphorinanes. Synthesis and Some Properties

Abstract

In contrast to the great variety of well known phosphorus heterocycles with exocyclic C=O double bond the number of ones containing exocyclic C=N bond is unusually modest. We elaborated the convenient method of preparation of various N,P,N-heterocycles with exocyclic C=N bond (1–4) from the readily available corresponding amino acid amidines and appropriated dichlorides or diamides of phosphorus (III) acids. Rings' 1–3 with P(III) are easily converted into (thio)phosphoryl derivatives, while the direct phosphorylation of amino amidines by RP(Y)Cl₂ is unusually ineffective. Tricycles 4 - derivatives of 2-(2-amino pheny1)imidazoline - mainly exist in the more conjugated hydrophosphazo tautomeric form 4b (>90 %).     

Irregular Boundary Value Problems with Discontinuous Coefficients and the Eigenvalue Parameter

In this study, a Birkhoff-irregular boundary value problem for linear ordinary differential equations of the second order with discontinuous coefficients and the spectral parameter has been considered. Therefore, at the discontinuous point, two additional boundary conditions (called transmission conditions) have been added to the boundary conditions. The eigenvalue parameter is of the second degree in the differential equation and of the first degree in a boundary condition. The equation contains an abstract linear operator which is (usually) unbounded in the space Lq(−1, 1). Isomorphism and coerciveness with defects 1 and 2 are proved for this problem. The case of the biharmonic equation is also studied.