Limit shape of probability measure on tensor product of BnB_n algebra modules

We study a probability measure on integral dominant weights in the decomposition of $N$-th tensor power of spinor representation of the Lie algebra $so(2n+1)$. The probability of the dominant weight $\lambda$ is defined as the ratio of the dimension of the irreducible component of $\lambda$ divided by the total dimension $2^{nN}$ of the tensor power. We prove that as $N\to \infty$ the measure weakly converges to the radial part of the $SO(2n+1)$-invariant measure on $so(2n+1)$ induced by the Killing form. Thus, we generalize Kerov's theorem for $su(n)$ to $so(2n+1)$.Comment: Submitted to Zapiski Nauchnykh Seminarov POM

Trapped modes in non-uniform elastic waveguides: asymptotic and numerical methods

Trapped modes within elastic waveguides are investigated employing asymptotic and numerical methods. The problems considered in this thesis concentrate on linear elastic waves in thickened/thinned and curved waveguides. The localised modes are propagating within some region that is characterized by a small parameter but are cut-off for geometric reasons exterior to that region, and thereafter exponentially decay with distance along the waveguide. Given this physical interpretation long wave theories become appropriate. The general approach is as follows: an asymptotic scheme is developed to analyse whether trapped modes should be expected and to obtain the frequencies at which trapped modes are excited. The asymptotic approach leads to an ordinary differential equation eigenvalue problem that encapsulates the essential physics. Then, numerical simulations based on spectral methods are performed for this reduced equation and for the full elasticity equations to validate the asymptotic scheme and demonstrate its accuracy. The thesis begins with an investigation of trapping due to thickness variations. The long-wave model for trapped modes is derived and it is shown that this model is functionally the same as that for a bent plate. Careful computations of the exact governing equations are compared with the asymptotic theory to demonstrate that the theories tie together. Different boundary conditions upon the guide walls and the importance of the sign of the group velocity are discussed in detail. Then, it is shown that boundary conditions also play a crucial role in the possible existence of trapped modes. The possibility of trapped modes is considered in nonuniform elastic/ ocean/ quantum waveguides where the guide has one wall with Dirichlet (clamped) boundary conditions and the other Neumann (stress-free) boundary conditions. For bent waveguides, with such boundary conditions, the sign of the curvature function is shown to play an important role in the possibility of trapping. Trapped modes in 3D elastic plates are considered as a model of waves that are guided along, and localised to the vicinity of, welds. These waves propagate unattenuated along the weld and exponentially decay with distance transverse to it. Three-dimensional geometries introduce additional complications but, again, asymptotic analysis is possible. The long-wave model provides numerical values of the trapped mode frequencies and gives conditions at which trapping can occur; these depend on the components of the wave number in different directions and variations of the plate thickness. To mimic the guide stretching out to infinity a perfectly matched layer (PML) technique originally developed by Berenger for electromagnetic wave propagation is employed. The method is illustrated on the example of topographically varying and bent acoustic guides, and numerically implemented in the spectral scheme to construct dispersion curves for a two-dimensional circular elastic annulus immersed in infinite fluid. This numerical scheme is new and more efficient than direct root-finding methods for the exact dispersion relation involving the Bessel functions. In the final chapter, the influence of external fluid on trapping within elastic waveguides is considered. A long-wave scheme for a curved and thickening plates in infinite fluid is derived, conditions of existence of trapping are analysed and compared with those for plates in vacuum

Tensor powers for non-simply laced Lie Algebras B2B_2 case

We study the decomposition problem for tensor powers of $B_2$-fundamental modules. To solve this problem singular weight technique and injection fan algorithms are applied. Properties of multiplicity coefficients are formulated in terms of multiplicity functions. These functions are constructed showing explicitly the dependence of multiplicity coefficients on the highest weight coordinates and the tensor power parameter. It is thus possible to study general properties of multiplicity coefficients for powers of the fundamental $B_2$- modules.Comment: 17 page

Diversity and noise effects in a model of homeostatic regulation of the sleep-wake cycle

Recent advances in sleep neurobiology have allowed development of physiologically based mathematical models of sleep regulation that account for the neuronal dynamics responsible for the regulation of sleep-wake cycles and allow detailed examination of the underlying mechanisms. Neuronal systems in general, and those involved in sleep regulation in particular, are noisy and heterogeneous by their nature. It has been shown in various systems that certain levels of noise and diversity can significantly improve signal encoding. However, these phenomena, especially the effects of diversity, are rarely considered in the models of sleep regulation. The present paper is focused on a neuron-based physiologically motivated model of sleep-wake cycles that proposes a novel mechanism of the homeostatic regulation of sleep based on the dynamics of a wake-promoting neuropeptide orexin. Here this model is generalized by the introduction of intrinsic diversity and noise in the orexin-producing neurons in order to study the effect of their presence on the sleep-wake cycle. A quantitative measure of the quality of a sleep-wake cycle is introduced and used to systematically study the generalized model for different levels of noise and diversity. The model is shown to exhibit a clear diversity-induced resonance: that is, the best wake-sleep cycle turns out to correspond to an intermediate level of diversity at the synapses of the orexin-producing neurons. On the other hand only a mild evidence of stochastic resonance is found when the level of noise is varied. These results show that disorder, especially in the form of quenched diversity, can be a key-element for an efficient or optimal functioning of the homeostatic regulation of the sleep-wake cycle. Furthermore, this study provides an example of constructive role of diversity in a neuronal system that can be extended beyond the system studied here.Comment: 18 pages, 12 figures, 1 tabl

A Physiologically Based Model of Orexinergic Stabilization of Sleep and Wake

The orexinergic neurons of the lateral hypothalamus (Orx) are essential for regulating sleep-wake dynamics, and their loss causes narcolepsy, a disorder characterized by severe instability of sleep and wake states. However, the mechanisms through which Orx stabilize sleep and wake are not well understood. In this work, an explanation of the stabilizing effects of Orx is presented using a quantitative model of important physiological connections between Orx and the sleep-wake switch. In addition to Orx and the sleep-wake switch, which is composed of mutually inhibitory wake-active monoaminergic neurons in brainstem and hypothalamus (MA) and the sleep-active ventrolateral preoptic neurons of the hypothalamus (VLPO), the model also includes the circadian and homeostatic sleep drives. It is shown that Orx stabilizes prolonged waking episodes via its excitatory input to MA and by relaying a circadian input to MA, thus sustaining MA firing activity during the circadian day. During sleep, both Orx and MA are inhibited by the VLPO, and the subsequent reduction in Orx input to the MA indirectly stabilizes sustained sleep episodes. Simulating a loss of Orx, the model produces dynamics resembling narcolepsy, including frequent transitions between states, reduced waking arousal levels, and a normal daily amount of total sleep. The model predicts a change in sleep timing with differences in orexin levels, with higher orexin levels delaying the normal sleep episode, suggesting that individual differences in Orx signaling may contribute to chronotype. Dynamics resembling sleep inertia also emerge from the model as a gradual sleep-to-wake transition on a timescale that varies with that of Orx dynamics. The quantitative, physiologically based model developed in this work thus provides a new explanation of how Orx stabilizes prolonged episodes of sleep and wake, and makes a range of experimentally testable predictions, including a role for Orx in chronotype and sleep inertia

Problems and contradictions of the financial system of the agro-business industry of Russia in contemporary conditions and methodological aspects of their overcoming

The article deals with the problems related to financial provision of investing activities of agribusiness industry as the most important sector of the Russian economy, as well as the ways of overcoming these problems. The authors formulated main concerns hindering economic growth and development of Russian agribusiness industry, and substantiated the dependence of the economic development of the agribusiness industry on the level, content, and mechanism of financial support. The article refines the major contradictions in the financial security of agribusiness industry developments, which reduces the level of financing and contributes to the emergence of disproportions in terms of both the industry-specific aspect and the territorial (regional) aspect in the development of the agricultural sector of the economy. It is proved that in spite of excessive government centralization of financial resources in the state budget, solutions to the critical national problems on the social arrangement of the agricultural regions and financial support of the needs of agribusiness industry are not achieved yet. The authors give critical assessment of financial and credit policy of the Bank of Russia, which led to the monopolization of the financial market and the emergence of speculative and often criminal banking operations to extract excessively high income. It is noted that wrong state policy has resulted in a significant differentiation in the development of Russian regions. The authors prove the necessity to align socio-economic development of agrarian regions in Russia and create their own food economy by ensuring the development of agricultural production with due considerations of climatic and other conditions characteristic of the various regions. The article reveals also causes and contradictions hindering financial well-being of enterprises of agro-industrial complex, substantiating and systemizing main objectives, methodological approaches, and organizational measures aimed at overcoming these contradictions in the development of financial support in Russia in the current context. Provision of financial resources in accordance with the proposed approaches and measures will allow comprehensively solving problems concerned both the development of infrastructure in rural areas, and peculiarly, the strengthening and growth of agricultural production.peer-reviewe

A mathematical model of sleep-wake cycles: the role of hypocretin/orexin in homeostatic regulation and thalamic synchronization

Sleep is vital to our health and well-being. Yet, we do not have answers to such fundamental questions as “why do we sleep?” and “what are the mechanisms of sleep regulation?”. Better understanding of these issues can open new perspectives not only in basic neurophysiology but also in different pathological conditions that are going along with sleep disorders and/or disturbances of sleep, e.g. in mental or neurological diseases. A generally accepted concept that explains regulation of sleep was proposed in 1982 by Alexander Borb´ely. It postulates that sleep-wake transitions result from the interaction between a circadian and a homeostatic sleep processes. The circadian process is ascribed to a “genetic clock” in the neurons of the suprachiasmatic nucleus of the hypothalamus. The mechanisms of the homeostatic process are still unclear. In this study a novel concept of hypocretin (orexin) - based control of sleep homeostasis is presented. The neuropeptide hypocretin is a synaptic co-transmitter of neurons in the lateral hypothalamus. It was discovered in 1998 independently by two different groups, therefore, obtaining two names, hypocretin and orexin. This neuropeptide is required to maintain wakefulness. Dysfunction in the hypocretin system leads to the sleep disorder narcolepsy, which, among other symptoms, is characterized by severe disturbances of sleep-wake cycles with sudden sleep-attacks in the wake period and interruptions of the sleep phase. On the other hand injection of hypocretin promotes wakefulness and improves the performance of sleep deprived subjects. The major proposals of the present study are the following: 1) the homeostatic regulation of sleep depends on the dynamics of a neuropeptide hypocretin; 2) ongoing impulse generation of the hypocretin neurons during wakefulness is sustained by reciprocal excitatory connections with other neurons, including local glutamate interneurons; 3) the transition to a silent state (sleep) is going along with an activity-dependent weakening of the hypocretin synaptic efficacy; 4) during the silent state (sleep) synaptic efficacy recovers and firing (wakefulness) can be reinstalled due to the circadian or other input. This concept is realized in a mathematical model of sleep-wake cycles which is built up on a physiology-based, although simplified Hodgkin-Huxley-type approach. In the proposed model a hypocretin neuron is reciprocally connected with a local interneuron via excitatory glutamate synapses. The hypocretin neuron additionally releases the neuropeptide hypocretin as co-transmitter. Besides of the local glutamate interneurons hypocretin neuron excites two gap junction coupled thalamic neurons. The functionally relevant changes are introduced via activity-dependent alterations of the synaptic efficacy of hypocretin. It is decreasing with each action potential generated by the hypocretin neuron. This effect is superimposed by a slow, continuous recovery process. The decreasing synaptic efficacy during the active wake state introduces an increasing sleep pressure. Ist dissipation during the silent sleep state results from the synaptic recovery. The model data demonstrate that the proposed mechanisms can account for typical alterations of homeostatic changes in sleep and wake states, including the effects of an alarm clock, napping and sleep deprivation. In combination with a circadian input, the model mimics the experimentally demonstrated transitions between different activity states of hypothalamic and thalamic neurons. In agreement with sleep-wake cycles, the activity of hypothalamic neurons changes from silence to firing, and the activity of thalamic neurons changes from synchronized bursting to unsynchronized single-spike discharges. These simulation results support the proposed concept of state-dependent alterations of hypocretin effects as an important homeostatic process in sleep-wake regulation, although additional mechanisms may be involved