The cubic period-distance relation for the Kater reversible pendulum

We describe the correct cubic relation between the mass configuration of a Kater reversible pendulum and its period of oscillation. From an analysis of its solutions we conclude that there could be as many as three distinct mass configurations for which the periods of small oscillations about the two pivots of the pendulum have the same value. We also discuss a real compound Kater pendulum that realizes this property.Comment: 25 pages 4figure

The use of the asymptotic expansion to speed up the computation of a series of spherical harmonics

When a function is expressed as an infinite series of spherical harmonics the convergence can be accelerated by subtracting its asymptotic expansion and adding it in analytically closed form. In the present article this technique is applied to two biophysical cases: to the potential distribution in a spherically symmetric volume conductor and to the covariance matrix of biomagnetic measurements

The use of the asymptotic expansion to speed up the computation of a series of spherical harmonics

When a function is expressed as an infinite series of spherical harmonics the convergence can be accelerated by subtracting its asymptotic expansion and adding it in analytically closed form. In the present article this technique is applied to two biophysical cases: to the potential distribution in a spherically symmetric volume conductor and to the covariance matrix of biomagnetic measurements

Sources of individual variation in plasma testosterone levels

The steroid hormone testosterone (T) plays a central role in the regulation of breeding in males, because many physiological, morphological and behavioural traits related to reproduction are T dependent. Moreover, in many seasonally breeding vertebrates, male plasma T levels typically show a pronounced peak during the breeding season. While such population-level patterns are fairly well worked out, the sources and the implications of the large variability in individual T levels within the seasonal cycle remain surprisingly little understood. Understanding the potential sources of individual variation in T levels is important for behavioural and evolutionary ecologists, for at least two reasons. First, in ‘honest signalling’ theory, T is hypothesized to play a critical role as the assumed factor that enforces honesty of the expression of sexually selected quality indicators. Second, T is often considered a key mediator of central life-history trade-offs, such as investment in survival versus reproduction or in mating versus parental care. Here, we discuss the patterns of within- and between-individual variation in male plasma T levels in free-living populations of birds. We argue that it is unclear whether this variability mainly reflects differences in underlying individual quality (intrinsic factors such as genetic or maternal effects) or in the environment (extrinsic factors including time of day, individual territorial status and past experience). Research in avian behavioural endocrinology has mainly focused on the effects of extrinsic factors, while other sources of variance are often ignored. We suggest that studies that use an integrative approach and investigate the relative importance of all potential sources of variation are essential for the interpretation of data on individual plasma T levels

Quantum Universe on Extremely Small Space-time Scales

The semiclassical approach to the quantum geometrodynamical model is used for the description of the properties of the Universe on extremely small space-time scales. Under this approach, the matter in the Universe has two components of the quantum nature which behave as antigravitating fluids. The first component does not vanish in the limit ħ → 0 and can be associated with dark energy. The second component is described by an extremely rigid equation of state and goes to zero after the transition to large space-time scales. On small space-time scales, this quantum correction turns out to be significant. It determines the geometry of the Universe near the initial cosmological singularity point. This geometry is conformal to a unit four-sphere embedded in a five-dimensional Euclidean flat space. During the consequent expansion of the Universe, when reaching the post-Planck era, the geometry of the Universe changes into that conformal to a unit four-hyperboloid in a five-dimensional Lorentz-signatured flat space. This agrees with the hypothesis about the possible change of geometry after the origin of the expanding Universe from the region near the initial singularity point. The origin of the Universe can be interpreted as a quantum transition of the system from a region in the phase space forbidden for the classical motion, but where a trajectory in imaginary time exists, into a region, where the equations of motion have the solution which describes the evolution of the Universe in real time. Near the boundary between two regions, from the side of real time, the Universe undergoes almost an exponential expansion which passes smoothly into the expansion under the action of radiation dominating over matter which is described by the standard cosmological model.Квазiкласичний пiдхiд до квантово-геометродинамiчної моделi застосовано для опису властивостей всесвiту на екстремально малих просторово-часових масштабах. У цьому пiдходi матерiя у всесвiтi має двi компоненти квантової природи, якi поводять себе як антигравiтуючi рiдини. Перша компонента не набуває нульового значення в границi ħ → 0 та може бути асоцiйована з темною енергiєю. Друга компонента описується екстремально жорстким рiвнянням стану i прямує до нуля пiсля переходу до великих просторово-часових масштабiв. На малих просторовочасових масштабах ця квантова поправка вiдiграє значну роль. Вона визначає геометрiю всесвiту бiля точки початкової космологiчної сингулярностi. Ця геометрiя є конформною до одиничної 4-сфери, зануреної у 5-вимiрний евклiдовий плоский простiр. Пiд час наступного розширення всесвiту, пiсля досягнення пост-планкiвської ери, геометрiя всесвiту перетворюється на геометрiю, конформну до одиничного 4-гiперболоїда у 5- вимiрному плоскому просторi з лоренцiвською сигнатурою. Це узгоджується з гiпотезою про можливу змiну геометрiї пiсля виникнення всесвiту, що розширюється з областi поблизу точки початкової сингулярностi. Виникнення всесвiту може бути iнтерпретовано як квантовий перехiд системи з областi у фазовому просторi, забороненої для класичного руху, але де iснує траєкторiя в уявному часi, в область, де рiвняння руху мають розв’язок, що описує еволюцiю всесвiту у реальному часi. Поблизу межi мiж двома областями, з боку реального часу, всесвiт зазнає майже експоненцiального розширення, яке гладко переходить у розширення пiд дiєю випромiнювання, що домiнує над матерiєю, у вiдповiдностi iз стандартною космологiчною моделлю

Peripheral and central inputs shape network dynamics in the developing visual cortex in vivo

Spontaneous network activity constitutes a central theme during the development of neuronal circuitry [1, 2]. Before the onset of vision, retinal neurons generate waves of spontaneous activity that are relayed along the ascending visual pathway [3, 4] and shape activity patterns in these regions [5, 6]. The spatiotemporal nature of retinal waves is required to establish precise functional maps in higher visual areas, and their disruption results in enlarged axonal projection areas (e.g., [7-10]). However, how retinal inputs shape network dynamics in the visual cortex on the cellular level is unknown. Using in vivo two-photon calcium imaging, we identified two independently occurring patterns of network activity in the mouse primary visual cortex (V1) before and at the onset of vision. Acute manipulations of spontaneous retinal activity revealed that one type of network activity largely originated in the retina and was characterized by low synchronicity (L-) events. In addition, we identified a type of high synchronicity (H-) events that required gap junction signaling but were independent of retinal input. Moreover, the patterns differed in wave progression and developmental profile. Our data suggest that different activity patterns have complementary functions during the formation of synaptic circuits in the developing visual cortex