High frequency homogenisation for elastic lattices

A complete methodology, based on a two-scale asymptotic approach, that enables the homogenisation of elastic lattices at non-zero frequencies is developed. Elastic lattices are distinguished from scalar lattices in that two or more types of coupled waves exist, even at low frequencies. Such a theory enables the determination of effective material properties at both low and high frequencies. The theoretical framework is developed for the propagation of waves through lattices of arbitrary geometry and dimension. The asymptotic approach provides a method through which the dispersive properties of lattices at frequencies near standing waves can be described; the theory accurately describes both the dispersion curves and the response of the lattice near the edges of the Brillouin zone. The leading order solution is expressed as a product between the standing wave solution and long-scale envelope functions that are eigensolutions of the homogenised partial differential equation. The general theory is supplemented by a pair of illustrative examples for two archetypal classes of two-dimensional elastic lattices. The efficiency of the asymptotic approach in accurately describing several interesting phenomena is demonstrated, including dynamic anisotropy and Dirac cones.Comment: 24 pages, 7 figure

ડૉ. બલભણુપ્રસાદ શાસ્ત્રીનાં રૂપકોનું સમીક્ષાત્મક અધ્યયન

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Geometrically navigating topological platonic modes around gentle and sharp bends

Predictive theory to geometrically engineer devices and materials in continuum systems to have desired topological-like effects is developed here by bridging the gap between quantum and continuum mechanical descriptions. A structured elastic plate, a bosoniclike system in the language of quantum mechanics, is shown to exhibit topological valley modes despite the system having no direct physical connection to quantum effects. We emphasize a predictive, first-principles, approach, the strength of which is demonstrated by the ability to design well-defined broadband edge states, resistant to backscatter, using geometric differences; the mechanism underlying energy transfer around gentle and sharp corners is described. Using perturbation methods and group theory, several distinct cases of symmetry-induced Dirac cones, which when gapped yield nontrivial band gaps, are identified and classified. The propagative behavior of the edge states around gentle or sharp bends depends strongly upon the symmetry class of the bulk media and we illustrate this via numerical simulations

High frequency homogenisation for elastic lattices

A complete methodology, based on a two-scale asymptotic approach, that enables the homogenisation of elastic lattices at non-zero frequencies is developed. Elastic lattices are distinguished from scalar lattices in that two or more types of coupled waves exist, even at low frequencies. Such a theory enables the determination of effective material properties at both low and high frequencies. The theoretical framework is developed for the propagation of waves through lattices of arbitrary geometry and dimension. The asymptotic approach provides a method through which the dispersive properties of lattices at frequencies near standing waves can be described; the theory accurately describes both the dispersion curves and the response of the lattice. The leading order solution is expressed as a product between the standing wave solution and the long-scale envelope functions that are eigensolutions of the homogenised partial differential equation. The general theory is supplemented by a pair of illustrative examples for two archetypal classes of two-dimensional elastic lattices. The efficiency of the asymptotic approach in accurately describing several interesting phenomena is demonstrated, including dynamic anisotropy and Dirac cones. A significant advantage of the method exposited herein is that it allows one to obtain analytical expressions for the leading order asymptotic solutions and effective material properties, even for complex systems; the methodology allows for greater physical understanding of the behaviour of the system than is usually the case for purely numerical homogenisation schemes

Asymptotic approximations for Bloch waves and topological mode steering in a planar array of Neumann scatterers

We study the canonical problem of wave scattering by periodic arrays, either of infinite or finite extent, of Neumann scatterers in the plane; the characteristic lengthscale of the scatterers is considered small relative to the lattice period. We utilise the method of matched asymptotic expansions, together with Fourier series representations, to create an efficient and accurate numerical approach for finding the dispersion curves associated with Floquet-Bloch waves through an infinite array of scatterers. The approach also lends itself to direct scattering problems for finite arrays and we illustrate the flexibility of these asymptotic representations on some topical examples from topological wave physics

Intracellular mediators of axonal sprouting in vivo.

Generation of new axonal sprouts and the process of axonal elongation play a vital role in neural regeneration and repair. The facial nerve axotomy model is a well-established, prototypical experimental paradigm for the systematic study of nerve regeneration and degeneration, providing insights into molecular signals that determine axonal regeneration and neuronal cell death. Interestingly, this model of peripheral injury induces a delayed appearance of galanin+ and calcitonin gene-related peptide+ (CGRP) neuropeptide-immunoreactive growth cones in the facial nucleus which peak at 14 days following axotomy and surprisingly, increase in number if recut within this time window. Furthermore, application of the retrograde tracer mini-ruby to the distal portion of the cut nerve demonstrates the motoneuron origin of these sprouting neurites. To gain an insight into the molecular mechanisms inducing the sprouting response we examined how neuronal cell death and the inflammatory response of various transgenics affected sprouting and regeneration. Deletion of the a7 integrin, which demonstrated a moderate reduction in regeneration, showed enhanced sprouting neural c-jun blocked regeneration, abolished regeneration-associated neuronal proteins and neuronal cell death, also completely eliminated central axonal sprouting. Absence of TNFR1&2 which displayed reduced neuronal cell death and inflammation, showed a tendency toward enhanced sprouting TGF(31 deletion, which showed an elevated inflammatory response and a 4-fold increase in neuronal cell death, resulted in decreased central sprouting. Similarly, enhanced neural inflammation following systemic injection of E.coli lipopolysaccharide (LPS) also reduced central sprouting. Finally, neuronally expressed constitutively active Ras (Ras+), dominant-negative MEK1 (MEKIdn) and Ras+xMEK1dn double mutant (DM) all demonstrated reduced neuronal cell death as well as substantially enhanced central sprouting, particularly in the MEKIdn mutant, suggesting that the sprouting response in these mutants may be beneficial for improving regeneration in the CNS. Sprouting and regeneration studies in Ras+ and MEKIdn mutants were therefore extended to the injured corticospinal tract (CST) and rubrospinal tract. These mutants showed extensive collateral sprouting of corticospinal tract (CST) axons, in the grey and white matter on the ipsilateral side in Ras+, MEKIdn and DM animals compared with wild-type (WT) controls when the injury spared the dorsolateral CST, enhanced green fluorescent protein (EGFP) labelled rubrospinal axons showed increased sprouting below the site of injury following a C4 injury in Ras+, MEKIdn and DM mice but this was not statistically significant compared with wild-type controls. To determine functional recovery rearing and grid-walk tests over 28 days following a unilateral left dorsal hemisection (DH) at C4 were used. Ras+, MEKIdn and DM groups performed significantly better in left forepaw use than WT in the rearing test at day 28 (25.0% 3.0% 32% 1% 50.0% 9.0%) compared with WT (13.0% 5.0%). Similarly, Ras, MEKIdn and DM animals showed significantly less footslips on the left forepaw compared with WT at day 28 (11.0% 2.0% 10.0% 1.0%+/- 12.0% 1.0% 19.0% 2.0% respectively). Overall, data from facial nucleus studies suggest central axonal sprouting is an injury but not a reinnervation-driven response that it is not directly connected to neuronal cell death, that excessive inflammation is detrimental, and that jun-, Ras-, and MEK1-mediated changes in regeneration-associated gene and protein expression play a vital part in shaping the growth cone response. Following spinal injury, expression of MEKIdn enhanced CST sprouting below the injury site. Furthermore, the combination with Ras+ also enhanced functional recovery following C4 DH. These data suggest that neuronal expression of active Ras and MEKIdn might serve as a promising biochemical strategy for regrowth in the injured spinal cord

Asymptotically exact photonic approximations of chiral symmetric topological tight-binding models

Topological photonic edge states, protected by chiral symmetry, are attractive for guiding wave energy as they can allow for more robust guiding and greater control of light than alternatives; however, for photonics, chiral symmetry is often broken by long-range interactions. We look to overcome this difficulty by exploiting the topology of networks, consisting of voids and narrow connecting channels, formed by the spaces between closely spaced perfect conductors. In the limit of low frequencies and narrow channels, these void-channel systems have a direct mapping to analogous discrete mass-spring systems in an asymptotically rigorous manner and therefore only have short-range interactions. We demonstrate that topological tight-binding models that are protected by chiral symmetries, such as the SSH model and square-root semimetals, are reproduced for these void-channel networks with appropriate boundary conditions. We anticipate, moving forward, that this paper provides a basis from which to explore continuum photonic topological systems, in an asymptotically exact manner, through the lens of a simplified tight-binding model

Hybrid topological guiding mechanisms for photonic crystal fibers

We create hybrid topological-photonic localisation of light by introducing concepts from the field of topological matter to that of photonic crystal fiber arrays. S-polarized obliquely propagating electromagnetic waves are guided by hexagonal, and square, lattice topological systems along an array of infinitely conducting fibers. The theory utilises perfectly periodic arrays that, in frequency space, have gapped Dirac cones producing band gaps demarcated by pronounced valleys locally imbued with a nonzero local topological quantity. These broken symmetry-induced stop-bands allow for localised guidance of electromagnetic edge-waves along the crystal fiber axis. Finite element simulations, complemented by asymptotic techniques, demonstrate the effectiveness of the proposed designs for localising energy in finite arrays in a robust manner

Endogenous transforming growth factor β1 suppresses inflammation and promotes survival in adult CNS

Transforming growth factor β1 (TGFβ1) is a pleiotropic cytokine with potent neurotrophic and immunosuppressive properties that is upregulated after injury, but also expressed in the normal nervous system. In the current study, we examined the regulation of TGFβ1 and the effects of TGFβ1 deletion on cellular response in the uninjured adult brain and in the injured and regenerating facial motor nucleus. To avoid lethal autoimmune inflammation within 3 weeks after birth in TGFβ1-deficient mice, this study was performed on a T- and B-cell-deficient RAG2-/- background. Compared with wild-type siblings, homozygous deletion of TGFβ1 resulted in an extensive inflammatory response in otherwise uninjured brain parenchyma. Astrocytes increased in GFAP and CD44 immunoreactivity; microglia showed proliferative activity, expression of phagocytosis-associated markers [αXβ2, B7.2, and MHC1 (major histocompatibility complex type 1)], and reduced branching. Ultrastructural analysis revealed focal blockade of axonal transport, perinodal damming of axonal organelles, focal demyelination, and myelin debris in granule-rich, phagocytic microglia. After facial axotomy, absence of TGFβ1 led to a fourfold increase in neuronal cell death (52 vs 13%), decreased central axonal sprouting, and significant delay in functional recovery. It also interfered with the microglial response, resulting in a diminished expression of early activation markers [ICAM1 (intercellular adhesion molecule 1), α6β1, and αMβ2] and reduced proliferation. In line with axonal and glial findings in the otherwise uninjured CNS, absence of endogenous TGFβ1 also caused an ∼10% reduction in the number of normal motoneurons, pointing to an ongoing and potent trophic role of this anti-inflammatory cytokine in the normal as well as in the injured brain. Copyright © 2007 Society for Neuroscience