B\"acklund Transformation for the BC-Type Toda Lattice

We study an integrable case of n-particle Toda lattice: open chain with boundary terms containing 4 parameters. For this model we construct a B\"acklund transformation and prove its basic properties: canonicity, commutativity and spectrality. The B\"acklund transformation can be also viewed as a discretized time dynamics. Two Lax matrices are used: of order 2 and of order 2n+2, which are mutually dual, sharing the same spectral curve.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Eigenproblem for Jacobi matrices: hypergeometric series solution

We study the perturbative power-series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d. The(small) expansion parameters are being the entries of the two diagonals of length d-1 sandwiching the principal diagonal, which gives the unperturbed spectrum. The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series of 3d-5 variables in the generic case, or 2d-3 variables for the eigenvalue growing from a corner matrix element. To derive the result, we first rewrite the spectral problem for a Jacobi matrix as an equivalent system of cubic equations, which are then resolved by the application of the multivariable Lagrange inversion formula. The corresponding Jacobi determinant is calculated explicitly. Explicit formulae are also found for any monomial composed of eigenvector's components.Comment: Latex, 20 pages; v2: corrected typos, added section with example

Dynamic characteristics of compliant journal bearings considering thermal effects

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Derived categories of Fano threefolds and degenerations

Using the technique of categorical absorption of singularities we prove that the nontrivial components of the derived categories of del Pezzo threefolds of degree $d \in \{2,3,4,5\}$ and crepant categorical resolutions of the nontrivial components of the derived categories of nodal del Pezzo threefolds of degree $d = 1$ can be smoothly deformed to the nontrivial components of the derived categories of prime Fano threefolds of genus $g = 2d + 2 \in \{4,6,8,10,12\}$. This corrects and proves the Fano threefolds conjecture of the first author from [Kuz09], and opens a way to interesting geometric applications, including a relation between the intermediate Jacobians and Hilbert schemes of curves of the above threefolds. We also describe a compactification of the moduli stack of prime Fano threefolds endowed with an appropriate exceptional bundle and its boundary component that corresponds to degenerations associated with del Pezzo threefolds.Comment: 40 page

Categorical absorptions of singularities and degenerations

We introduce the notion of categorical absorption of singularities: an operation that removes from the derived category of a singular variety a small admissible subcategory responsible for singularity and leaves a smooth and proper category. We construct (under appropriate assumptions) a categorical absorption for a projective variety $X$ with isolated ordinary double points. We further show that for any smoothing $\mathcal{X}/B$ of $X$ over a smooth curve $B$, the smooth part of the derived category of $X$ extends to a smooth and proper over $B$ family of triangulated subcategories in the fibers of $\mathcal{X}$.Comment: 39 pages; v2: minor improvement

Q-operator and factorised separation chain for Jack polynomials

Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P(x_1,...,x_n) are eigenfunctions of a one-parameter family of integral operators Q_z. The operators Q_z are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q_{z_k} we construct an integral operator S_n factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S_n admits a factorisation described in terms of restricted Jack polynomials P(x_1,...,x_k,1,...,1). Using the operator Q_z for z=0 we give a simple derivation of a previously known integral representation for Jack polynomials.Comment: 26 page

How, why and to what extent are sustainability disclosures produced in the Norwegian O&G offshore supply companies?

Masteroppgave i Sustainable management - Høgskolen i Bodø, 2010This research paper makes an atternpt to find out the answers to questions of the extent, reasons and mechanisms of 'sustainability disclosures production in Norway within the O&G offshore supply companies, the cluster of the Norwegian Shipowners Association. The problem of sustainability disclosures and accountability of business seems interesting for the detail ed exploration, taking into account the specific contextual peculiarities of the chosen industry. Nowadays the issues of sustainable management and the disclosure sustainability data on the special indicators are considered as quite new dimensions in the process of managing an industrial organization. The O&G industry and its offshore supply cluster in Norway have always been under the widespread attention of different interested groups of society; that is why the issue of sustainability disclosures as a tool to reduce uncertainties between business and society is worth being analyzed in-depth. In order to answer the research question we have accomplished the following tasks. At first, we have represented the operationalization of the central concepts in our work - 'sustainability' and 'sustainability disclosures' from both theoretical and practical points of view. The tas k of defining the extent to which the Norwegian offshore supply companies produce sustainability disclosures has been carried out through a survey of the list of companies, which has shown the distribution of standards and guidelines applied to produce such disclosures basing on the commitment to the national and international legislation and the voluntaryapproaches. The tasks of reasons and mechanisms of sustainability disclosures have being done through the analysis of case studies of two offshore suppliers - Acergy and Technip Norge, which understand and disclose data on sustainability in their own different ways. The results of our research have shown that the sustainability concept of sustainability is operationalized through the lens of 'project engineering' context of the O&G offshore supply operations. The main accent is made on the strategic objectives of quality, health, safety, environment and security within the engineering daily routines in the offshore. The motivation for the regular production of sustainability disclosures bases on the pressure of the most crucial stakeholder groups: the national government, clients, and the employees unions, every of which has the particular data expectations. The mechanisms of sustainability disclosures are characterized by the domination of internal procedural standards and the data disclosure only to particular stakeholders. The framework of international sustainability reporting standards application is represented weak in the offshore supply cluster, but has a potential for development

Modal logic of planar polygons

We study the modal logic of the closure algebra $P_2$, generated by the set of all polygons in the Euclidean plane $\mathbb{R}^2$. We show that this logic is finitely axiomatizable, is complete with respect to the class of frames we call "crown" frames, is not first order definable, does not have the Craig interpolation property, and its validity problem is PSPACE-complete