Metabelian SL(n,C) representations of knot groups IV: twisted Alexander polynomials

In this paper we will study properties of twisted Alexander polynomials of knots corresponding to metabelian representations. In particular we answer a question of Wada about the twisted Alexander polynomial associated to the tensor product of two representations, and we settle several conjectures of Hirasawa and Murasugi

Splittings of knot groups

Let K be a knot of genus g. If K is fibered, then it is well known that the knot group pi(K) splits only over a free group of rank 2g. We show that if K is not fibered, then pi(K) splits over non-free groups of arbitrarily large rank. Furthermore, if K is not fibered, then pi(K) splits over every free group of rank at least 2g. However, pi(K) cannot split over a group of rank less than 2g. The last statement is proved using the recent results of Agol, Przytycki-Wise and Wise.Comment: 28 pages, 2 figure

TOWARDS A FINANCIALLY OPTIMAL DESIGN OF IT SERVICES

The current financial crisis forces companies to allocate IT budgets more effectively and thus increases the demand for suitable methods to evaluate the financial impact of IT investments. This especially applies to service-orientation, a design paradigm which facilitates the standardisation and flexibilisation of business processes and IT applications, topics that currently are very much in vogue in science and practice. This paper focuses on the realisation of a new functionality by IT services and presents a methodology to determine their financially optimal functional scope on the continuum between realising just one IT service providing the whole functionality and realising many IT services each providing only a small share of functionality. This approach allows for a multi-period financial valuation of an uncertain demand for the new functionality, as well as of an uncertain company-wide reuse of the corresponding IT services. Finally, the methodology is evaluated by an example from a financial services provider

High rate locally-correctable and locally-testable codes with sub-polynomial query complexity

In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length $n$, constant rate (which can even be taken arbitrarily close to 1), constant relative distance, and query complexity $\exp(\tilde{O}(\sqrt{\log n}))$. Previously such codes were known to exist only with $\Omega(n^{\beta})$ query complexity (for constant $\beta > 0$), and there were several, quite different, constructions known. Our codes are based on a general distance-amplification method of Alon and Luby~\cite{AL96_codes}. We show that this method interacts well with local correctors and testers, and obtain our main results by applying it to suitably constructed LCCs and LTCs in the non-standard regime of \emph{sub-constant relative distance}. Along the way, we also construct LCCs and LTCs over large alphabets, with the same query complexity $\exp(\tilde{O}(\sqrt{\log n}))$, which additionally have the property of approaching the Singleton bound: they have almost the best-possible relationship between their rate and distance. This has the surprising consequence that asking for a large alphabet error-correcting code to further be an LCC or LTC with $\exp(\tilde{O}(\sqrt{\log n}))$ query complexity does not require any sacrifice in terms of rate and distance! Such a result was previously not known for any $o(n)$ query complexity. Our results on LCCs also immediately give locally-decodable codes (LDCs) with the same parameters

Mechanism-based model characterizing bidirectional interaction between PEGylated liposomal CKD-602 (S-CKD602) and monocytes in cancer patients

S-CKD602 is a PEGylated liposomal formulation of CKD-602, a potent topoisomerase I inhibitor. The objective of this study was to characterize the bidirectional pharmacokinetic-pharmacodynamic (PK-PD) interaction between S-CKD602 and monocytes. Plasma concentrations of encapsulated CKD-602 and monocytes counts from 45 patients with solid tumors were collected following intravenous administration of S-CKD602 in the phase I study. The PK-PD models were developed and fit simultaneously to the PK-PD data, using NONMEM®. The monocytopenia after administration of S-CKD602 was described by direct toxicity to monocytes in a mechanism-based model, and by direct toxicity to progenitor cells in bone marrow in a myelosuppression-based model. The nonlinear PK disposition of S-CKD602 was described by linear degradation and irreversible binding to monocytes in the mechanism-based model, and Michaelis-Menten kinetics in the myelosuppression-based model. The mechanism-based PK-PD model characterized the nonlinear PK disposition, and the bidirectional PK-PD interaction between S-CKD602 and monocytes. © 2012 Cárdenas et al, publisher and licensee Dove Medical Press Ltd

The Turaev and Thurston norms

In 1986, W. Thurston introduced a (possibly degenerate) norm on the first cohomology group of a 3-manifold. Inspired by this definition, Turaev introduced in 2002 a analogous norm on the first cohomology group of a finite 2-complex. We show that if N is the exterior of a link in a rational homology sphere, then the Thurston norm agrees with a suitable variation of Turaev's norm defined on any 2-skeleton of N.Comment: 17 pages. V2: We deleted one direction of Lemma 4.5 since the proof was incorrect. This does not affect any of the other results of the pape

The Grothendieck group of polytopes and norms

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The Optimal Single Copy Measurement for the Hidden Subgroup Problem

The optimization of measurements for the state distinction problem has recently been applied to the theory of quantum algorithms with considerable successes, including efficient new quantum algorithms for the non-abelian hidden subgroup problem. Previous work has identified the optimal single copy measurement for the hidden subgroup problem over abelian groups as well as for the non-abelian problem in the setting where the subgroups are restricted to be all conjugate to each other. Here we describe the optimal single copy measurement for the hidden subgroup problem when all of the subgroups of the group are given with equal a priori probability. The optimal measurement is seen to be a hybrid of the two previously discovered single copy optimal measurements for the hidden subgroup problem.Comment: 8 pages. Error in main proof fixe