Two new time-dependent versions of div-curl results in a bounded domain \domain⊂\RR3 are presented. We study a limit of the product \vectorvk\vectorwk, where the sequences \vectorvk and \vectorwk belong to \Lp2. In Theorem ??? we assume that \rotor\vectorvk is bounded in the Lp-norm and \diver\vectorwk is controlled in the Lr-norm. In Theorem ??? we suppose that \rotor\vectorwk is bounded in the Lp-norm and \diver\vectorwk is controlled in the Lr-norm. The time derivative of \vectorwk is bounded in both cases in the norm of \Hk−1. The convergence (in the sense of distributions) of \vectorvk\vectorwk to the product \vectorv\vectorw of weak limits of \vectorvk and \vectorwk is shown.
Sağlam (2004) investigates the influence of foreign words on Turkish, presenting the extent of borowings from different source languages, known as the "etymological spectrum". The spectrum is considered to be the result of a diversification process. The present paper demonstrates that the etymological spectrum of Turkish abides by a general model that Altmann (1993) deduced for any linguistic rank orders.
In a recent paper (Diversity in Monoids, Czech. Math. J. 62 (2012), 795–809), the last two authors introduced and developed the monoid invariant “diversity” and related properties “homogeneity” and “strong homogeneity”. We investigate these properties within the context of inside factorial monoids, in which the diversity of an element counts the number of its different almost primary components. Inside factorial monoids are characterized via diversity and strong homogeneity. A new invariant complementary to diversity, height, is introduced. These two invariants are connected with the well-known invariant of elasticity.
Let $M$ be a (commutative cancellative) monoid. A nonunit element $q\in M$ is called almost primary if for all $a,b\in M$, $q\mid ab$ implies that there exists $k\in \mathbb {N}$ such that $q\mid a^k$ or $q\mid b^k$. We introduce a new monoid invariant, diversity, which generalizes this almost primary property. This invariant is developed and contextualized with other monoid invariants. It naturally leads to two additional properties (homogeneity and strong homogeneity) that measure how far an almost primary element is from being primary. Finally, as an application the authors consider factorizations into almost primary elements, which generalizes the established notion of factorization into primary elements.
Firing properties of single neurons in the nervous system have been recognized to be determined by their intrinsic ion channel dynamics and extrinsic synaptic inputs. Previous studies have suggested that dendritic structures exhibit significant roles in the modulation of somatic firing behavior in neurons. Following these studies, we show that finite information transmission delay between dendrite and soma can also influence the somatic firings in neurons. Our investigation is based on a two-compartment model which can approximately reproduce the firing activity of cortical pyramidal neurons. The obtained simulation results indicate that under subthreshold stimulus, spontaneous fast spiking activity is induced by large values of time delay, while for suprathreshold stimulus, regular bursting, chaotic firing and fast spiking can be observed under different time delays. More importantly, the transition mode between these diverse firing patterns with the variation of delay shows a period-doubling phenomenon under certain stimulus intensity. Consequently, our model results can not only illustrate the influential roles of internal time delay in the generation of a diversity of neuronal firing patterns, but also provide us with frameworks for investigating the impacts of internal time delay on the firing properties of many other neurons in the nervous system.