5,6‐Di­phenyl­[1,3]­di­thiolo­[4,5‐b]di­thiine‐2‐thione

In the title compound, C₁₇H₁₀S₅, the dithiine ring adopts a boat conformation while the di­thiole ring has an envelope conformation. The phenyl groups are planar and make dihedral angles of 40.7 (2) and 59.8 (2)° with the best plane of the thiine ring. The shortest intermolecular S?S contact is 3.305 (2) Å.     

Multirectangular characteristic invariants for power ?-Köthe spaces of first type

Let ? be a Banach sequence space with a monotone norm ‖⋅?_‖, in which the canonical system (ei) is a normalized unconditional basis. We consider the problem of quasi-diagonal isomorphism of first type power ?-Köthe spaces E?(λ,a) (see (1) below). From [P.A. Chalov, V.P. Zahariuta, On quasi-diagonal isomorphism of generalized power spaces, in: Linear Topological Spaces and Complex Analysis, vol. 2, METU - TÜB?TAK, Ankara, 1995, pp. 35-44; P.A. Chalov, T. Terzio?lu, V.P. Zahariuta, First type power Köthe spaces and m-rectangular invariants, in: Linear Topological Spaces and Complex Analysis, vol. 3, METU - TÜB?TAK, Ankara, 1997, pp. 30-44; P.A. Chalov, T. Terzio?lu, V.P. Zahariuta, Multirectangular invariants for power Köthe spaces, J. Math. Anal. Appl. 297 (2004) 673-695] it is known that the system of all m-rectangle characteristics μm (see (9) below) is a complete quasi-diagonal invariant on the class of all first type power Köthe spaces [V.P. Zahariuta, On isomorphisms and quasi-equivalence of bases of power Köthe spaces, Soviet Math. Dokl. 16 (1975) 411-414; V.P. Zahariuta, Linear topologic invariants and their applications to isomorphic classification of generalized power spaces, Turkish J. Math. 20 (1996) 237-289], if the relation of equivalency of systems and is defined by some natural estimates with constants independent of m. Deriving the characteristic from the characteristic β (see [V.P. Zahariuta, Linear topological invariants and isomorphisms of spaces of analytic functions, in: Matem. Analiz i ego Pril., vol. 2, Rostov Univ., Rostov-on-Don, 1970, pp. 3-13 (in Russian), in: Matem. Analiz i ego Pril., vol. 3, Rostov Univ., Rostov-on-Don, 1971, pp. 176-180 (in Russian); V.P. Zahariuta, Generalized Mityagin invariants and a continuum of mutually nonisomorphic spaces of analytic functions, Funktsional. Anal. i Prilozhen. 11 (1977) 24-30 (in Russian); V.P. Zahariuta, Compact operators and isomorphisms of Köthe spaces, in: Aktualnye Voprosy Matem. Analiza, vol. 46, Rostov Univ., Rostov-on-Don, 1978, pp. 62-71 (in Russian); P.A. Chalov, P.B. Djakov, V.P. Zahariuta, Compound invariants and embeddings of Cartesian products, Studia Math. 137 (1) (1999) 33-47; P.B. Djakov, M. Yurdakul, V.P. Zahariuta, Isomorphic classification of Cartesian products, Michigan Math. J. 43 (1996) 221-229; V.P. Zahariuta, Linear topologic invariants and their applications to isomorphic classification of generalized power spaces, Turkish J. Math. 20 (1996) 237-289], and using the S. Krein's interpolation method of analytic scale, we are able to generalize some results of [P.A. Chalov, V.P. Zahariuta, On quasi-diagonal isomorphism of generalized power spaces, in: Linear Topological Spaces and Complex Analysis, vol. 2, METU - TÜB?TAK, Ankara, 1995, pp. 35-44; P.A. Chalov, T. Terzio?lu, V.P. Zahariuta, First type power Köthe spaces and m-rectangular invariants, in: Linear Topological Spaces and Complex Analysis, vol. 3, METU - TÜB?TAK, Ankara, 1997, pp. 30-44; P.A. Chalov, T. Terzio?lu, V.P. Zahariuta, Multirectangular invariants for power Köthe spaces, J. Math. Anal. Appl. 297 (2004) 673-695].     

More on θ-compact fuzzy topological spaces

Recently, El-Naschie has shown that the notion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and ε^∞ theory. In 2005, Caldas and Jafari have introduced θ-compact fuzzy topological spaces. The purpose of this paper is to investigate further properties of θ-compact fuzzy topological spaces. Moreover, the notion of θ-closed fuzzy topological spaces is introduced and properties of it are obtained.     

Difference equations with a point interaction

In this paper, we consider an impulsive second‐order difference equation on the whole axis. We determine eigenvalues, spectral singularities, continuous spectrum corresponding to this difference equation with an impulsive condition by using the asymptotic properties of Jost functions, and uniqueness theorems of analytic functions. Finally, we demonstrate that the impulsive difference equation has finite number of eigenvalues and spectral singularities with finite multiplicities under certain conditions.     

FPTAS for half-products minimization with scheduling applications

A special class of quadratic pseudo-boolean functions called “half-products” (HP) has recently been introduced. It has been shown that HP minimization, while NP-hard, admits a fully polynomial time approximation scheme (FPTAS). In this note, we provide a more efficient FPTAS. We further show how an FPTAS can also be derived for the general case where the HP function is augmented by a problem-dependent constant and can justifiably be assumed to be nonnegative. This leads to an FPTAS for certain partitioning type problems, including many from the field of scheduling.     

Robust scheduling and robustness measures for the discrete time/cost trade-off problem

Projects are often subject to various sources of uncertainties that have a negative impact on activity durations and costs. Therefore, it is crucial to develop effective approaches to generate robust project schedules that are less vulnerable to disruptions caused by uncontrollable factors. In this paper, we investigate the robust discrete time/cost trade-off problem, which is a multi-mode project scheduling problem with important practical relevance. We introduce surrogate measures that aim at providing an accurate estimate of the schedule robustness. The pertinence of each proposed measure is assessed through computational experiments. Using the insights revealed by the computational study, we propose a two-stage robust scheduling algorithm. Finally, we provide evidence that the proposed approach can be extended to solve a complex robust problem with tardiness penalties and earliness revenues.     

Generalized submaximal spaces

The notion of generalized submaximal spaces is introduced and studied. Properties and characterizations of generalized submaximal spaces are discussed.     

Nuclear spallation as a mechanism for radioisotope production: Cross sections for selected nuclides

Proton spallation is a relatively recent method for significant radioisotope production. A detailed discussion of the activation equation for charged-particle reactions is given, and measured values of spallation cross sections for reactions of 800-MeV protons with three target materials are reported.     

Intramolecular interactions in ethyl bis-(2,4-dinitrophenyl)acetate

The crystal structure of ethyl bis(2,4-dinitrophenyl)acetate, determined by room temperature X-ray diffraction, shows that this crowded molecule contains two short intramolecular interactions between nitro oxygen atoms and electron deficient carbon atoms. One mimics an early stage of the nucleophilic addition to a carbonyl group (O...C, 2.730 Å), and the other mimics an early stage of a Michael reaction (O...C, 2.854 Å). Crystal data: C₁₆H₁₂N₄O₁₀, M _r = 420.29, a = 10.079(1), b = 11.139(1), c = 16.162(1) Å, = 91.319(7)°, monoclinic, P2₁/c, Z = 4.