Elastic shallow shells with functionally graded structure

Main objects of considerations are linear-elastic thin shallow shells with a tolerance-periodic structure along one direction. Moreover, their properties along the perpendicular direction can be described by continuous, slowly-varying functions. Hence, these objects can be treated as shells with a functionally graded structure. Here, the tolerance model of these shells is proposed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control-oriented Rotordynamics

The rotor vibrations are coupled by many parameters: angular speed couples two radial directions, unbalance couples torsional and lateral vibrations, rotor support couples rigid body and flexible modes. Small changes of these parameters strongly influence the rotor dynamics. The full analysis of the rotor coupled vibrations is much simpler when we can divide the system into smaller subsystems. In this case the calculations are simplified and we have deep insight into mechanisms leading to good or bed behaviour of the rotor motion. It is particularly important in the case of the rotor working in the wide range of the angular speeds. For the vibration analysis the methods known from control theory can be applied. The proposed approach was testified in the paper on the simple 3-mode rotor model. The torsional vibrations were separated from the lateral vibrations and a feedback among the subsystems was established. The subsystems are coupled by rotor unbalance and root locus method allows to show the critical values of the unbalance which destabilize rotor motion for different angular speeds. The lateral vibrations are stabilized by angular speed and again it is possible to find how big value of the rotor speed is sufficient to stabilize the rotor motion. Such analysis of the rotor vibrations appeared very useful for the choice of the control strategy. It indicated such control laws which amplify the stabilizing mechanisms in the rotor dynamics. Such procedures can also lead to the energy saving control laws. In the case of the lateral vibrations there were considered four control strategies. And these strategies were compared to indicate optimal one. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Remarks on the r and Δ convolutions

In this paper we study the properties of the r–deformation introduced in [B1]. We observe that the associated convolution coming from the conditionally free convolution is associative only for r = 1 and r = 0. We give the realization of some r–Gaussian random variables and obtain Haagerup–Pisier–Buchholz type inequalities. We also study another convolution defined with the use of the r–deformation through a moment–cumulant formula [KY1] and show that it is associative and in general not positive. Partially sponsored with KBN grant no 2P03A00723 and RTN HPRN-CT-2002-00279.     

Interpolation of Bilinear Operators Between Quasi-Banach Spaces

We study interpolation, generated by an abstract method of means, of bilinear operators between quasi-Banach spaces. It is shown that under suitable conditions on the type of these spaces and the boundedness of the classical convolution operator between the corresponding quasi-Banach sequence spaces, bilinear interpolation is possible. Applications to the classical real method spaces, Calderón-Lozanovsky spaces, and Lorentz-Zygmund spaces are presented. The author is supported by the National Science Foundation under grant DMS 0099881. The author is supported by KBN Grant 1 P03A 013 26.     

Some results on Vizing’s conjecture and related problems

The well-known conjecture of Vizing on the domination number of Cartesian product graphs claims that for any two graphs $G$ and $H$, $\gamma \left(G\square H\right)\ge \gamma \left(G\right)\gamma \left(H\right)$. We disprove its variations on independent domination number and Barcalkin–German number, i.e. Conjectures 9.6 and 9.2 from the recent survey Bre?ar et al. (2012) [4]. We also give some extensions of the double-projection argument of Clark and Suen (2000) [8], showing that their result can be improved in the case of bounded-degree graphs. Similarly, for rainbow domination number we show for every $k\ge 1$ that ${\gamma }_{rk}\left(G\square H\right)\ge \frac{k}{k+1}\gamma \left(G\right)\gamma \left(H\right)$, which is closely related to Question 9.9 from the same survey. We also prove that the minimum possible counterexample to Vizing’s conjecture cannot have two neighboring vertices of degree two.     

An Extension of the Birkhoff Integrability for Multifunctions

A comparison between a set-valued Gould type and simple Birkhoff integrals of bf(X)-valued multifunctions with respect to a non-negative set function is given. Relationships among them and Mc Shane multivalued integrability is given under suitable assumptions.     

$$L^{p}$$-Norms a nd Mahler’s Measure of Poly nomials o n the n-Dime nsio nal Torus

We prove Nikol’skii type inequalities that, for polynomials on the n-dimensional torus \(\mathbb {T}^n\), relate the \(L^p\)-norm with the \(L^q\)-norm (with respect to the normalized Lebesgue measure and \(0 <p <q < \infty \)). Among other things, we show that \(C=\sqrt{q/p}\) is the best constant such that \(\Vert P\Vert _{L^q}\le C^{\text {deg}(P)} \Vert P\Vert _{L^p}\) for all homogeneous polynomials P on \(\mathbb {T}^n\). We also prove an exact inequality between the \(L^p\)-norm of a polynomial P on \(\mathbb {T}^n\) and its Mahler measure M(P), which is the geometric mean of |P| with respect to the normalized Lebesgue measure on \(\mathbb {T}^n\). Using extrapolation, we transfer this estimate into a Khintchine–Kahane type inequality, which, for polynomials on \(\mathbb {T}^n\), relates a certain exponential Orlicz norm and Mahler’s measure. Applications are given, including some interpolation estimates.