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1
Content available remote Continuous Pairwise Comparisons
Saaty T. L.
Fundamenta Informaticae
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2016
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Vol. 144, nr 3/4
213--221
EN
One often assumes that comparisons are discrete and carried out in a matrix of numbers. However, our eyes and other senses perform comparisons in a continuous way by making many simultaneously. Here the mathematics of pairwise comparisons is generalized to the continuous case. It is more likely that in evolution, little creatures had and now still have a "feeling" about the environment in which they find themselves by sensing many things at once. Through evolution, sensing has been transformed to thinking and to discrete comparisons. This paper presents some material from other works written about continuous comparisons.
2
Content available Variational characterizations for eigenfunctions of analytic self-adjoint operator functions
Katsouleas G., Maroulas J.
Opuscula Mathematica
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2013
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Vol. 33, no. 2
307--321
EN
In this paper we consider Rellich's diagonalization theorem for analytic self-adjoint operator functions and investigate variational principles for their eigenfunctions and interlacing statements. As an application, we present a characterization for the eigenvalues of hyperbolic operator polynomials.
3
Content available Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type
Iakovlev S. I., Iakovleva V.
Opuscula Mathematica
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2013
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Vol. 33, no. 1
81-98
EN
It is shown that any μ ∈ C is an infinite multiplicity eigenvalue of the Steklov smoothing operator Sh acting on the space [formula]. For μ ≠ 0 the eigenvalue-eigenfunction problem leads to studying a differential-difference equation of mixed type. An existence and uniqueness theorem is proved for this equation. Further a transformation group is defined on a countably normed space of initial functions and the spectrum of the generator of this group is studied. Some possible generalizations are pointed out.
4
Content available remote The dynamic stability of a simply supported stepped beam with additional discrete elements
Sochacki W.
Vibrations in Physical Systems
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2012
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Vol. 25
367--372
EN
The dynamic stability of a simply supported stepped beam with additional discrete elements was investigated in the paper. These elements are a rotational spring and a rotary inertia, both of which are connected to the beam. The discrete elements can be mounted at any chosen position along the beam length. The influence of step changes in the cross-section of the beam on its dynamic stability was also investigated in the paper. The problem of dynamic stability was solved by applying the mode summation method. Applying an orthogonal condition of eigenfunctions, the dynamic of the system was described with the use of the Mathieu equation. The obtained equation allowed the dynamic stability of the tested system to be analysed. The considered beam was treated as Euler-Bernoulli beam.
5
Content available remote The dynamic stability of beams with step changes in cross-section
Sochacki W.
Vibrations in Physical Systems
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2008
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Vol. 23
327--333
EN
The influence of step changes in cross-section of beams with different boundary conditions on their dynamic stability was investigated in the paper. The change in the crosssections took place in an optional location along the beam length. The investigated beams were axially loaded by a force in the form P(t)= P0+Scosνt. The problem of dynamic stability was solved by applying the mode summation method. The obtained Mathieu equation allowed the dynamic stability of tested systems to be analysed. The analysis relied on testing the influence of step changes in beam cross-sections and their locations on the value of coefficient b in the Mathieu equation. The considered beams were treated as Euler- Bernoulli beams.
6
Content available remote The regularity of measures and related properties of eigenfunctions and first eigenvalues of some fractal elliptic operators
Triebel H.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2004
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Vol. 44, nr spec.
257-283
EN
The paper deals with regularity properties of potentials of Radon measures u in IR^n, expressed in terms of some fractal quantities of u (Theorem 1). Based on these assertions, first eigenvalues and eigenfunctions of some fractal elliptic operators are considered (Theorems 2 and 3). The results are illustrated by examples.
7
Design of output feedback compensator for smart structure model via reduced order model
Umapathy M., Bandyopadhyay B., Unbehauen H.
Systems Science
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2002
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Vol. 28, nr 4
61-84
EN
A method of designing an output feedback compensator for vibration control of a flexible smart cantilever beam based on its reduced order model is presented. By retaining the first two vibration modes the state space model is obtained from a smart structure Finite Element Model (FEM). A reduced order model is obtained by retaining the first vibration mode. It has been shown that an output feedback compensator can be obtained for the smart structure model from the state feedback gains designed from its reduced order model. It has also been shown that if the compensator is placed in the closed loop with the higher order system, it guarantees the closed loop stability. As the states are not needed for feedback, the method is simple and can be easily implemented.
8
Content available remote Effective measurements of birefringence properties of nondichroic media using Poincare sphere.
Kurzynowski P., Ratajczyk F.
Optica Applicata
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2001
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Vol. 31, nr 1
203-207
EN
A method for measuring the birefringence properties of nondichroic media using the Poincare sphere is presented. Simple relations between coordinates of points on the Poincare sphere representing input and output polarization states of light and the point representing first eigenvector of the medium have been found. From these relations the desired polarization parameters of the medium were calculated.
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