Minimal antiderivatives and monotonicity

We consider settings in convex analysis which give rise to families of convex functions that contain their lower envelope. Given certain partial data regarding a subdifferential, we consider the family of all convex antiderivatives that comply with the given data. We prove that this family is not empty and, in particular, contains a minimal antiderivative under a fairly general assumption on the given data. It turns out that the representation of monotone operators by convex functions fits naturally in these settings. Duality properties of representing functions are also captured by these settings, and the gap between the Fitzpatrick function and the Fitzpatrick family is filled by this broader sense of minimality of the Fitzpatrick function.     

Novel coumarin derivatives containing 1,2,4-triazole, 4,5-dicyanoimidazole and purine moieties: synthesis and evaluation of their cytostatic activity

We report here on the synthesis and in vitro anti-tumor effects of a series of novel 1,2,4-triazole (compounds 3-6), 4,5-dicyanoimidazole (compound 7), and purine (compounds 8-13) coumarin derivatives and their acyclic nucleoside analogues 14-18. Structures of novel compounds 3-18 were deduced from their ¹H- and ¹³C-NMR and corresponding mass spectra. Results of anti-proliferative assays performed on a panel of selected human tumor cell lines revealed that compound 6 had moderate cytostatic activity against the HeLa cell line (IC₅₀ = 35 μM), whereas compound 10 showed moderate activity against the HeLa (IC₅₀ = 33 μM), HepG2 (IC₅₀ = 25 μM) and SW620 (IC₅₀ = 35 μM) cell lines. These compounds showed no cytotoxic effects on normal (diploid) human fibroblasts.     

Conical averagedness and convergence analysis of fixed point algorithms

Journal of Global Optimization - We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of...     

Efficient palladium-mediated or base-induced 5-endo-dig cyclisation of C5-alkynylated pyrimidine derivatives: conventional and microwave-assisted synthesis of novel furo[2,3-d]pyrimidines

A series of the novel 5-alkynyl- and furo[2,3-d]pyrimidine derivatives in which the sugar moiety is replaced by a methoxymethyl (MOM) group is synthesised using the Sonogashira cross-coupling reaction under both conventional and microwave conditions, in good to excellent yields. The 5-endo-dig cyclisation of 5-alkynylpyrimidine derivatives promoted by a Pd-catalyst or base gives the corresponding furo[2,3-d]pyrimidines in good yields.     

Optimal Pricing for Optimal Transport

Suppose that c(x, y) is the cost of transporting a unit of mass from x ∈ X to y ∈ Y and suppose that a mass distribution μ on X is transported optimally (so that the total cost of transportation is minimal) to the mass distribution ν on Y. Then, roughly speaking, the Kantorovich duality theorem asserts that there is a price f(x) for a unit of mass sold (say by the producer to the distributor) at x and a price g(y) for a unit of mass sold (say by the distributor to the end consumer) at y such that for any x ∈ X and y ∈ Y, the price difference g(y) ? f(x) is not greater than the cost of transportation c(x, y) and such that there is equality g(y) ? f(x) = c(x, y) if indeed a nonzero mass was transported (via the optimal transportation plan) from x to y. We consider the following optimal pricing problem: suppose that a new pricing policy is to be determined while keeping a part of the optimal transportation plan fixed and, in addition, some prices at the sources of this part are also kept fixed. From the producers’ side, what would then be the highest compatible pricing policy possible? From the consumers’ side, what would then be the lowest compatible pricing policy possible? We have recently introduced and studied settings in c-convexity theory which gave rise to families of c-convex c-antiderivatives, and, in particular, we established the existence of optimal c-convex c-antiderivatives and explicit constructions of these optimizers were presented. In applications, it has turned out that this is a unifying language for phenomena in analysis which used to be considered quite apart. In the present paper we employ optimal c-convex c-antiderivatives and conclude that these are natural solutions to the optimal pricing problems mentioned above. This type of problems drew attention in the past and existence results were previously established in the case where X = Y = ? ⁿ under various specifications. We solve the above problem for general spaces X, Y and real-valued, lower semicontinuous cost functions c. Furthermore, an explicit construction of solutions to the general problem is presented.     

Eco-friendly synthesis,in vitro anti-proliferative evaluation,and 3D-QSAR analysis of a novel series of monocationic 2-aryl/heteroaryl-substituted 6-(2-imidazolinyl)benzothiazole mesylates

Abstract

Herein, we describe the synthesis of twenty-one novel water-soluble monocationic 2-aryl/heteroaryl-substituted 6-(2-imidazolinyl)benzothiazole mesylates 3a–3u and present the results of their anti-proliferative assays. Efficient syntheses were achieved by three complementary simple two-step synthetic protocols based on the condensation reaction of aryl/heteroaryl carbaldehydes or carboxylic acid. We developed an eco-friendly synthetic protocol using glycerol as green solvent, particularly appropriate for the condensation of thermally and acid-sensitive heterocycles such as furan, benzofuran, pyrrole, and indole. Screening of anti-proliferative activity was performed on four human tumour cell lines in vitro including pancreatic cancer (CFPAC-1), metastatic colon cancer (SW620), hepatocellular carcinoma (HepG2), and cervical cancer (HeLa), as well as in normal human fibroblast cell lines. All tested compounds showed strong to moderate anti-proliferative activity on tested cell lines depending on the structure containing aryl/heteroaryl moiety coupled to 6-(2-imidazolinyl)benzothiazole moiety. The most potent cytostatic effects on all tested cell lines with \(\hbox {IC}_{50}\) values ranging from 0.1 to 3.70 \(\upmu \hbox {M}\) were observed for benzothiazoles substituted with naphthalene-2-yl 3c, benzofuran-2-yl 3e, indole-3-yl 3j, indole-2-yl 3k, quinoline-2-yl 3s, and quinoline-3-yl 3t and derivatives substituted with phenyl 3a, naphthalene-1-yl 3b, benzothiazole-2-yl 3g, benzothiazole-6-yl 3h, N-methylindole-3-yl 3l, benzimidazole-2-yl 3n, benzimidazole-5(6)-yl 3o, and quinolone-4-yl 3u with \(\hbox {IC}_{50}\) values ranging from 1.1 to 29.1 \(\upmu \hbox {M}\). Based on obtained anti-proliferative activities, 3D-QSAR models for five cell lines were derived. Molecular volume, molecular surface, the sum of hydrophobic surface areas, molecular mass, and possibility of making dispersion forces were identified by QSAR analyses as molecular properties that are positively correlated with anti-proliferative activity, while compound’s capability to accept H-bond was identified as a negatively correlated property. Comparison of molecular properties identified for different cell lines enabled assumptions about similarity of mode of action through which anti-proliferative activities against different cell lines are accomplished. Novel compounds that are predicted to have enhanced activities in comparison with herein presented ones were designed using 3D-QSAR analysis as guideline.

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Fitzpatrick functions,cyclic monotonicity and Rockafellar’s antiderivative

Several deeper results on maximal monotone operators have recently found simpler proofs using Fitzpatrick functions. In this paper, we study a sequence of Fitzpatrick functions associated with a monotone operator. The first term of this sequence coincides with the original Fitzpatrick function, and the other terms turn out to be useful for the identification and characterization of cyclic monotonicity properties. It is shown that for any maximal cyclically monotone operator, the pointwise supremum of the sequence of Fitzpatrick functions is closely related to Rockafellar’s antiderivative. Several examples are explicitly computed for the purpose of illustration. In contrast to Rockafellar’s result, a maximal 3-cyclically monotone operator need not be maximal monotone. A simplified proof of Asplund’s observation that the rotation in the Euclidean plane by π/n$\pi /n$ is n$n$-cyclically monotone but not (n+1)$(n+1)$-cyclically monotone is provided. The Fitzpatrick family of the subdifferential operator of a sublinear and of an indicator function is studied in detail. We conclude with a new proof of Moreau’s result concerning the convexity of the set of proximal mappings.     

Abstract convex optimal antiderivatives

Having studied families of antiderivatives and their envelopes in the setting of classical convex analysis, we now extend and apply these notions and results in settings of abstract convex analysis. Given partial data regarding a c-subdifferential, we consider the set of all c-convex c-antiderivatives that comply with the given data. Under a certain assumption, this set is not empty and contains both its lower and upper envelopes. We represent these optimal antiderivatives by explicit formulae. Some well known functions are, in fact, optimal c-convex c-antiderivatives. In one application, we point out a natural minimality property of the Fitzpatrick function of a c-monotone mapping, namely that it is a minimal antiderivative. In another application, in metric spaces, a constrained Lipschitz extension problem fits naturally the convexity notions we discuss here. It turns out that the optimal Lipschitz extensions are precisely the optimal antiderivatives. This approach yields explicit formulae for these extensions, the most particular case of which recovers the well known extensions due to McShane and Whitney.