An overview on available models and techniques for the multicriteria reliability problem with emphasis on a potential use in a DSS for this problem type

The paper reviews available models and techniques covering three classes of the multi-criteria reliability problem. These classes include allocation of reliabilities in presence of multiple goals, reliability optimization with several kinds of failures and allocation of resources to optimize multiattribute systems. Optimization techniques are also suggested. The models and techniques are also evaluated with respect to their potential use in a DSS for reliability optimization.     

Aromatic and Antiaromatic Pathways in Triphyrin(2.1.1) Annelated with Benzo[b]heterocycles

Understanding of the aromatic properties and magnetically induced current densities of highly conjugated chromophores is important when designing molecules with strongly delocalized electronic structure. Linear extension of the triphyrin(2.1.1) skeleton with an annelated benzo[b]heterocycle fragment modifies the aromatic character by extending the electron delocalization pathway. Two-electron reduction leads to an antiaromatic triphyrin(2.1.1) ring and an aromatic benzo[b]heterocycle subunit. Current-density calculations provide detailed information about the observed pathways and their strengths.     

Numerical optimisation of chemotherapy dosage under antiangiogenic treatment in the presence of drug resistance

We consider a two-compartment model of chemotherapy resistant tumour growth under angiogenic signalling. Our model is based on the one proposed by Hahnfeldt et al. (1999), but we divide tumour cells into sensitive and resistant subpopulations. We study the influence of antiangiogenic treatment in combination with chemotherapy. The main goal is to investigate how sensitive are the theoretically optimal protocols to changes in parameters quantifying the interactions between tumour cells in the sensitive and resistant compartments, that is, the competition coefficients and mutation rates, and whether inclusion of an antiangiogenic treatment affects these results. Global existence and positivity of solutions and bifurcations (including bistability and hysteresis) with respect to the chemotherapy dose are studied. We assume that the antiangiogenic agents are supplied indefinitely and at a constant rate. Two optimisation problems are then considered. In the first problem a constant, indefinite chemotherapy dose is optimised to maximise the time needed for the tumour to reach a critical (fatal) volume. It is shown that maximum survival time is generally obtained for intermediate drug dose. Moreover, the competition coefficients have a more visible influence on survival time than the mutation rates. In the second problem, an optimal dosage over a short, 30-day time period, is found. A novel, explicit running penalty for drug resistance is included in the objective functional. It is concluded that, after an initial full-dose interval, an administration of intermediate dose is optimal over a broad range of parameters. Moreover, mutation rates play an important role in deciding which short-term protocol is optimal. These results are independent of whether antiangiogenic treatment is applied or not.     

SHAH: SHape-Adaptive Haar Wavelets for Image Processing

We propose the shape-adaptive Haar (SHAH) transform for images, which results in an orthonormal, adaptive decomposition of the image into Haar-wavelet-like components, arranged hierarchically according to decreasing importance, whose shapes reflect the features present in the image. The decomposition is as sparse as it can be for piecewise-constant images. It is performed via a stepwise bottom-up algorithm with quadratic computational complexity; however, nearly linear variants also exist. SHAH is rapidly invertible. We show how to use SHAH for image denoising. Having performed the SHAH transform, the coefficients are hard- or soft-thresholded, and the inverse transform taken. The SHAH image denoising algorithm compares favorably to the state of the art for piecewise-constant images. A clear asset of the methodology is its very general scope: it can be used with any images or more generally with any data that can be represented as graphs or networks.     

Isoperimetric Inequalities and Imbedding Theorems in Irregular Domains

The paper proves several weighted imbedding theorems for domainswith fractal boundaries. The weights considered are distancesto the boundary to certain powers, and the domains are so-calleds-John domains. The paper also proves, in the general setting,that the existence of an imbedding implies compactness of theimbedding for lower exponents. Moreover, following Maz'ya, thepaper reformulates the imbedding theorems in the language oflocal isoperimetric and capacity estimates.     

Knots,BPS States,and Algebraic Curves

We analyze relations between BPS degeneracies related to Labastida-Mariño-Ooguri-Vafa (LMOV) invariants and algebraic curves associated to knots. We introduce a new class of such curves, which we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture, which is stronger than the known M-theory integrality predictions. Furthermore, we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally, we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.     

On a problem of Rolewicz about Banach spaces that admit support sets

We construct an example of a nonseparable Banach space which does not admit a support set.² It is a consistent (and necessarily independent from the axioms of ZFC) example of a space C(K) of continuous functions on a compact Hausdorff K with the supremum norm. The construction depends on a construction of a Boolean algebra with some combinatorial properties. The space is also hereditarily Lindelöf in the weak topology but it doesn't have any nonseparable subspace nor any nonseparable quotient which is a C(K) space for K dispersed.