Isolated Singularities of Polyharmonic Operator in Even Dimension

We consider the equation $\Delta^2 u=g(x,u) \geq 0$ in the sense of distribution in $\Omega'=\Omega\setminus \{0\}$ where $u$ and $-\Delta u\geq 0.$ Then it is known that $u$ solves $\Delta^2 u=g(x,u)+\alpha \delta_0-\beta \Delta \delta_0,$ for some non-negative constants $\alpha$ and $\beta.$ In this paper we study the existence of singular solutions to $\Delta^2 u= a(x) f(u)+\alpha \delta_0-\beta \Delta \delta_0$ in a domain $\Omega\subset \mathbb{R}^4,$ $a$ is a non-negative measurable function in some Lebesgue space. If $\Delta^2 u=a(x)f(u)$ in $\Omega',$ then we find the growth of the nonlinearity $f$ that determines $\alpha$ and $\beta$ to be $0.$ In case when $\alpha=\beta =0,$ we will establish regularity results when $f(t)\leq C e^{\gamma t},$ for some $C, \gamma>0.$ This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions $(N\geq 5)$ with a specific weight function $a(x)=|x|^\sigma.$ Later we discuss its analogous generalization for the polyharmonic operator

Cohomology of Lie algebroid over Algebraic spaces

We consider Lie algebroid over an algebraic space as a quasicoherent sheaf of Lie-Rinehart algebras. We compute algebraic (analytic) de Rham cohomologies for some free divisors and the associated logarithmic de Rham cohomologies as well. We express hypercohomology for a locally free Lie algebroid (of finite or infinte rank) as derived functor and simplify it via $\check{C}$ech cohomology. Furthermore, we define the Hochschild hypercohomology of a sheaf of generalized bialgebras and study the special cases, namely Hochschild hypercohomology of universal enveloping algebroid and jet algebroid of a Lie algebroid. We present a version of Hochschild-Kostant-Rosenberg (HKR) theorem for a locally free Lie algebroid as well as its dual version.Comment: Major revision has been don

Shakespeare, Macbeth and the Hindu Nationalism of Nineteenth-Century Bengal

The essay examines a Bengali adaptation of Macbeth, namely Rudrapal Natak (published 1874) by Haralal Ray, juxtaposing it with differently accented commentaries on the play arising from the English-educated elites of 19th Bengal, and relating the play to the complex phenomenon of Hindu nationalism. This play remarkably translocates the mythos and ethos of Shakespeare’s original onto a Hindu field of signifiers, reformulating Shakespeare’s Witches as bhairavis (female hermits of a Tantric cult) who indulge unchallenged in ghastly rituals. It also tries to associate the gratuitous violence of the play with the fanciful yearning for a martial ideal of nation-building that formed a strand of the Hindu revivalist imaginary. If the depiction of the Witch-figures in Rudrapal undercuts the evocation of a monolithic and urbane Hindu sensibility that would be consistent with colonial modernity, the celebration of their violence may be read as an effort to emphasize the inclusivity (as well as autonomy) of the Hindu tradition and to defy the homogenizing expectations of Western enlightenmen

Hermitian Lie algebroids over analytic spaces

We consider some aspects of complex Riemannian geometry for complex algebraic varieties and study Hermitian metrics on analytic spaces. Then we define Hermitian metrics on a holomorphic Lie algebroid and consider the associated characteristic foliation with canonically induced inner product. Moreover, we consider an example of a Hermitian Lie algebroid $\mathcal{L}$ and describe the induced inner product on a special $\mathcal{L}$-invariant subspace. Later, we consider hypercohomologies associated with leaf spaces, leaves and some $\mathcal{L}$-invariant subspaces for the characteristic foliation $\mathfrak{a}(\mathcal{L})$ of a holomorphic Lie algebroid $\mathfrak{a}: \mathcal{L} \rightarrow \mathcal{T}_X$ over a Hermitian manifold $X$

Multiphysics analysis of electrochemical and electromagnetic system addressing lithium-ion battery and permanent magnet motor

Lithium-ion batteries are the leading energy storage technology in the electronic-driven society. With the need for portable, long-life electronics the demand for lithium batteries has escalated over the decade. Lithium-ion batteries show remarkable electrochemical characteristics, including but not limited to, long cycle-life, high cut-off voltages and high energy-density. However, lithium-ion cells are problematic to design due to their inherent thermal and/or mechanical instability. The objective of the current research framework is to establish the criteria causing thermo-mechanical failure of the battery systems, material properties effecting the performance, and model cycle-life degradation due to electrolyte loss by solid electrolyte interface (SEI) formation. An extension of this thermo-mechanical analysis was performed on electromagnetic system. A FEM was performed for a 20W BLDC motor to predict the electromagnetic and thermo-mechanical performance under steady state operating conditions. In our present research, we have studied the mechanical and thermal aspect of lithium battery electrodes. The first and second project encapsulated the material selection aspect for thermo-mechanically stable lithium battery electrodes. The objective of these projects was to develop a set of material indices (five for mechanical and five for thermal) which compare the performance of electrode materials based on heat generation, diffusion and mechanical strength and toughness. A mathematical model was formulated to determine particle deformation and stress fields based upon an elastic-perfectly plastic constitutive response. Mechanical deformation was computed by combining the stress equilibrium equations with the electrochemical diffusion of lithium ions into the electrode particle. The result provided a time developing stress field which shifts from purely elastic to partially plastic deformation as the lithium-ion diffuses into the particle. For the mechanical integrity, the materials were tested for strength, and toughness under elastic and plastic deformation. The model was used to derive five merit indices that parametrize mechanical stability of electrode materials. The five indices were used to analyze the mechanical stability for the six candidate electrode materials – graphite, silicon, and titanium oxide for the anode and lithium manganese oxide, lithium cobalt oxide and lithium ferrous phosphate for the cathode. Finally, the work suggested ways to improve the mechanical performance of electrode materials and helps to identify mechanical and design properties that need to be considered for optimal electrode material selection. Materials were selected based upon high strength and toughness with the ability to handle faster charging capabilities. A coupled thermo-chemical model was developed and used for deriving the heat generation by electrode particle of different materials. The thermal merit index analysis was based on performing a multivariable material selection based on four mechanisms of thermal generation against the thermal diffusion characteristics of the electrode material. A new mode of heat generation was conceptualized plausible for fast charging electrode materials. The heat generated by this mechanism accounted for the strain energy dissipated due to plastic deformation of the electrode particles upon lithiation. A parametric analysis was conducted to compare the thermal performance of six candidate electrode materials (for cathode and anode) using the merit indices and the results were validated against past experimental data. The effect of variable charging rates on thermal generation was analyzed. Finally, the paper identified the material properties which affect the thermal performance of battery systems. The thermo-mechanical material indices were designed to be a tool or platform for industries and experimentalists to compare new with existing electrode materials and isolate the material properties that need to be altered for better performance of the battery. The third project undertaken was to work on the concept of structurally integrable and mechanically robust lithium-ion pouch cells applicable for hybrid electric vehicles. Branching out from the focus area of flexible lithium batteries, a structurally stable cell could be integrated with the body of the vehicle, thereby eliminating the additional weight and support needed to install a battery pack. The analysis involves the conceptualization of the lithium-ion separator membrane as an open-cell foam under compression and the decrement in the ionic-conductivity was modelled analogous to the permeability loss in a foamy material. The thermal profiling for three different lithium-ion cells (LCO/C, (lithium manganese oxide) LMO/C and LFP/C) were simulated with five separator materials under variable applied load, rates of charging and cooling conditions. A set of thermal maps were created to demarcate the domains of thermal meltdown of the separator membrane and the conditions leading up to the thermal runaway. The proposed model could be used as a design tool for industrial application of structurally flexible lithium-ion pouch cell to predict thermally safe lithium battery, thereby reducing the risks and loss from battery meltdown during prototyping. The fourth project undertook the modeling of battery degradation and life prediction due to SEI growth resulting into capacity fading. An efficient reduced-order electrochemical model was developed for lithium cobalt oxide (LCO)/graphite (C) pouch cell and a reaction-diffusion based SEI model was integrated to predict the cyclic capacity loss due to electrolyte deposition over the anode in the form of SEI. The experimental data was fitted based on a single-parameter fit to predict the reaction coefficient for SEI current. The algorithm developed for this battery module was designed to reduce the computational time for capacity fade calculation. The model was also applied for a lithium ferrous phosphate (LFP)/C cell without any fitting, and in both cases the predictions were within ±1% deviation from the experimental results, thereby predicting capacity fading for different cathode materials with graphite as the anode. A novel concept was developed in which “aged-battery” could be used as an advantage for biomedical and EV applications. The fading rate decays as the cell ages and aged-cells could be operated for longer life cycles with negligible fading. A cost analysis was performed to find the optimized point where the benefits from lower fading was weighed against the cost (material and electricity) involved in ageing the cell. The application for this concept would be in biomedical and EV industries, where the replacement of lithium-ion batteries over short periods of time is not feasible and the cost/risk of replacement exceeds the cost of ageing the battery. Aging the cell could prolong its cycle life thereby reducing the chances of battery replacement in a long duration of operation. The fifth project undertook a small project to compare the prediction of thermal conductivity by different approaches of Boltzmann Transport Equations. The lattice thermal conductivity predictions for a silicon nanoparticle was performed using three popular formulations of the Boltzmann transport equation. The models as proposed by Klemens, Callaway and Holland, essentially differ in the phonon scattering mechanisms and the vibrational modes considered in the respective formulations. At low temperatures, results from all three models showed strong agreement with experimental measurements but deviated significantly with increasing temperatures. Estimates from the Holland model, which explicitly accounted for the normal and Umklapp scattering processes of the transverse and longitudinal modes, concur with the measured values. Similar predictions were obtained from both Holland and Callaway models at high temperatures since phonon transport was dominated by longitudinal modes, as revealed from our analyses of the relaxation times. In conclusion, the paper inferred the importance of mode dependent thermal conduction in silicon nanoparticle at elevated temperatures. The final work done was to model a 20W BLDC motor with bonded magnets used as the surface permanent magnet for the rotor. A thermo-mechanical and electromagnetic analysis was performed to test the application of the 65 vol.% bonded NdFeB magnets in a motor. The performance analysis involved the prediction thermo-mechanical properties for the bonded magnets and redesign of the motor to operate at safe thermal and mechanical limits. The design was finalized and considered for prototyping as a part of the demonstration for the project. In conclusion, a thermo-mechanical multiphysics analysis and material selection was performed primarily for electrochemical and extended to electromagnetic systems to predict performance, mechanism of degradation and cycle life under variable operating conditions. These model act as tools and design guide to aid in the development of lithium-ion batteries and electromagnetic drives. The purpose of modeling and material selection is to reduce the cost of experimentation and prototyping prior to commercialization. The multiphysics modeling performed also isolates the parameters which effect the health and safety of the system, thereby reducing the risks of failure during operation. Therefore, selection of the correct design parameters and models to support the performance and life predictions allow a rapid and economic transition from prototyping to commercialization of electrochemical and electromagnetic systems