Matrix polynomials play an important role in the theory of matrix differential equations. We develop a fixed point method to compute solutions of matrix polynomials equations, where the matricial elements of the matrix polynomial are considered separately as complex polynomials. Numerical examples illustrate the method presented.
For a $C^1$-function $f$ on the unit ball $\mathbb B \subset \mathbb C ^n$ we define the Bloch norm by $\|f\|_\mathfrak B=\sup \|\tilde df\|,$ where $\tilde df$ is the invariant derivative of $f,$ and then show that $$ \|f\|_\mathfrak B= \sup _{z,w\in {\mathbb B} \atop z\neq w} (1-|z|^2)^{1/2}(1-|w|^2)^{1/2}\frac {|f(z)-f(w)|}{|w-P_wz-s_wQ_wz|}.$$.
† Pyrenicocephalus jarzembowskii, gen. et sp. n. (Hemiptera: Heteroptera: Enicocephalomorpha: Enicocephalidae: Enicocephalinae) from Early Eocene, London Clay, England, Isle of Sheppey, is described and illustrated according to the unique pyritized adult head reported as a larval enicocephalid head by Jarzembowski (1986). The head anatomy of similar and related genera of Enicocephalinae is compared and the close relationship of the new genus to a clade including the extant genera Oncylocotis, Embolorrhinus and Hoplitocoris is suggested, most probably as the sister genus to Hoplitocoris (presently with Afrotropical, East Palaearctic and Oriental range).
We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $\rho $-integral, introduced by Jarník and Kurzweil. Let $(\mathcal S_{\rho } (E), \Vert \cdot \Vert )$ be the space of all strongly $\rho $-integrable functions on a multidimensional compact interval $E$, equipped with the Alexiewicz norm $\Vert \cdot \Vert $. We show that each element in the dual space of $(\mathcal S_{\rho } (E), \Vert \cdot \Vert )$ can be represented as a strong $\rho $-integral. Consequently, we prove that $fg$ is strongly $\rho $-integrable on $E$ for each strongly $\rho $-integrable function $f$ if and only if $g$ is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on $E$.
An infinite series which arises in certain applications of the Lagrange-Bürmann formula to exponential functions is investigated. Several very exact estimates for the Laplace transform and higher moments of this function are developed.
In elementary robotics, it is very well known that the rotation of an object by the angles respectively Ψ (x), Θ (y), Φ (z) wrt** a fixed coordinate system (RPY) results in the same angular position for the object as the position achieved by the rotation of that object by the angles respectively Φ (z), Θ (y), Ψ (x) wrt a moving (with the object) coordinate system (euler angles). The proofs given up to now for such consequences are not general and for any such problem usually involve the calculation of the transformation matrix for both cases and observing the equivalence of the two matrices [1, 2, 3]. In this paper a fundamental and at the same time general proof is given for such results. It is shown that this equivalence in reverse order can be extended to the general class of transformations which keep the local relations constant (i.e., each transformation should keep the local relations constant). For example, rotation, translation and scaling are 3 types of transformations which can be located in this general class.
The paper presents an application of fuzzy logic modeling techniques for design and development of a classification system for car driver's vigilance level detection. Especially, the micro-sleeps detection is of our primary interest. Detection is based on a pattern analysis of EEG signal spectrograms, which are acquired by monitoring the driver during a driving process. The system is based on the concept of radial implicative fuzzy system, which can be treated as a logical system accommodating acquired knowledge in a structured form.