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

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

On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras

In this paper we study the asymptotic of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character probability measures. We derive the asymptotic distribution of irreducible components for the Plancherel measure, generalizing results of Biane and Tate and Zelditch. We also derive the asymptotic of the character measure for generic parameters and an intermediate scaling in the vicinity of the Plancherel measure. It is interesting that the asymptotic measure is universal and after suitable renormalization does not depend on which representations were multiplied but depends significantly on the degeneracy of the parameter in the character distribution.Comment: 24 pages; minor corrections, references adde

Post-COVID junior physics lab: The new normal

Physics laboratory is the most challenging aspect of teaching physics in a pandemic environment: How can we teach experimental skills when students are not in the lab? How do we ensure that both on-campus and online students develop relevant experimental skills and enjoy labs? Finally, how do we ensure COVID safety when students work in groups? In junior physics labs there is an additional challenge of scaling-up any teaching approach to large student cohorts. At the School of Physics, we teach cohorts of ~800 students per semester over four units of study at different levels: fundamental, regular and advanced. In this presentation we will share our experience and lessons learned over the last three-four years moving from teaching labs in the pre-pandemic world to the current new normal that includes both on-campus and online labs with hundreds of students in each stream.   Back in 2019, our Junior Physics Labs were very traditional: printed lab manuals, hand-written logbooks, bench notes as supportive materials, crowded classes, hand-drawn graphs, in-person paper tests, etc. We just moved into a new beautiful lab space and had been working on modifying lab curriculum, as well as lab equipment which had been largely unchanged for 20 years. However, in early 2020 the COVID-19 pandemic forced universities, including The University of Sydney (USYD), to move all classes online. For us this happened right at the start of semester, so it was necessary to quickly find a way to run labs in an online format. This included both running the experiments and managing all assignments, groupwork, and logbooks online. After some trial and error (including hybrid) over 2020-2021, we have set up completely independent online labs which now run in parallel with the campus labs and receive good feedback from remote students. They also provide a fallback plan for students who are in COVID-19 isolation and cannot attend the labs in-person. This transition also required a new approach to labs navigation on Canvas (web-based learning management system used by USYD) so we designed and developed new pages for both on-campus and online streams so that students can easily find required information and materials for each week. We introduced e-Lab manuals, shared e-logbooks, online quizzes, practical online tests, videos, and simulators which are now used in both in-person and online labs, elevating student experience and simplifying lab coordination and management. The main software tools that we use are Canvas, Zoom, and MS Office 365 (or Google Docs/Sheets). In addition to these we use mobile apps, e.g. Phyphox, and simulators, e.g. MultiSim and Phet, for doing or simulating experiments at home. Fast forward to 2022, physics labs at The University of Sydney have been returned to the fully face-to-face mode, though many students are still overseas. It is likely that many will also prefer to study remotely in the longer-term. In this presentation we discuss the rationale behind incorporating features of online labs into face-to-face labs and discuss how to run engaging and fun labs while maintaining appropriate social distancing and hygiene standards

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

Skew Howe duality and limit shapes of Young diagrams

We consider the skew Howe duality for the action of certain dual pairs of Lie groups $(G_1, G_2)$ on the exterior algebra $\bigwedge(\mathbb{C}^{n} \otimes \mathbb{C}^{k})$ as a probability measure on Young diagrams by the decomposition into the sum of irreducible representations. We prove a combinatorial version of this skew Howe for the pairs $(\mathrm{GL}_{n}, \mathrm{GL}_{k})$, $(\mathrm{SO}_{2n+1}, \mathrm{Pin}_{2k})$, $(\mathrm{Sp}_{2n}, \mathrm{Sp}_{2k})$, and $(\mathrm{Or}_{2n}, \mathrm{SO}_{k})$ using crystal bases, which allows us to interpret the skew Howe duality as a natural consequence of lattice paths on lozenge tilings of certain partial hexagonal domains. The $G_1$-representation multiplicity is given as a determinant formula using the Lindstr\"om-Gessel-Viennot lemma and as a product formula using Dodgson condensation. These admit natural $q$-analogs that we show equals the $q$-dimension of a $G_2$-representation (up to an overall factor of $q$), giving a refined version of the combinatorial skew Howe duality. Using these product formulas (at $q =1$), we take the infinite rank limit and prove the diagrams converge uniformly to the limit shape.Comment: 54 pages, 12 figures, 2 tables; v2 fixed typos in Theorem 4.10, 4.14, shorter proof of Theorem 4.6 (thanks to C. Krattenthaler), proved of Conjecture 4.17 in v