This book contains the proceedings of the conference on Compact Moduli and Vector Bundles, held from October 21-24, 2010, at the University of Georgia. This book is a mix of survey papers and ...original research articles on two related subjects: Compact Moduli spaces of algebraic varieties, including of higher-dimensional stable varieties and pairs, and Vector Bundles on such compact moduli spaces, including the conformal block bundles. These bundles originated in the 1970s in physics; the celebrated Verlinde formula computes their ranks. Among the surveys are those that examine compact moduli spaces of surfaces of general type and others that concern the GIT constructions of log canonical models of moduli of stable curves. The original research articles include, among others, papers on a formula for the Chern classes of conformal classes of conformal block bundles on the moduli spaces of stable curves, on Looijenga's conjectures, on algebraic and tropical Brill-Noether theory, on Green's conjecture, on rigid curves on moduli of curves, and on Steiner surfaces.
We consider the pull-back of a natural sequence of cohomology classes Θg,n∈H2(2g−2+n)(M‾g,n,Q) to the moduli space of stable maps M‾g,n(P1,d). These classes are related to the Brézin-Gross-Witten tau ...function of the KdV hierarchy via ZBGW(ħ,t0,t1,...)=exp∑ħ2g−2n!∫M‾g,nΘg,n⋅∏j=1nψjkj∏tkj. Insertions of the pull-backs of the classes Θg,n into the integrals defining Gromov-Witten invariants define new invariants which we show in the case of target P1 are given by a random matrix integral and satisfy the Toda equation.
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We define the analogue of instanton sheaves on the blow-up Pn˜ of the n-dimensional projective space at a point. We choose an appropriate polarisation on Pn˜ and construct rank 2 examples of locally ...free and non locally free (but torsion free) type. In general, the defined instantons also turn out to be the cohomology of monads, although non-linear ones. Moreover, in the five dimensional case, we show that there are continuous families of them that fill, at least, a smooth component in the moduli of semi-stable sheaves.
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•Two synchronous measurement methods of different moduli for asphalt mixture were proposed.•The effectiveness of two synchronous measurement methods was verified.•The correlation and ...difference of moduli obtained by four test methods were analyzed.
Two new test methods were proposed to measure the different moduli of asphalt mixtures simultaneously based on four-point bending and indirect tensile tests. The new calculating formulas were derived for tensile and compressive moduli of asphalt mixtures under four-point bending loading model, which combining the equilibrium condition and the plane hypothesis in elastic mechanics theory. Meanwhile, on the basis of the test principle of indirect tensile moduli and Hooke’s law in two-dimensional stress states, the new calculating formulas were derived in indirect tensile loading model. The moduli tests of four-point bending, indirect tensile, direct tension and unconfined compression were carried out separately to verify effectiveness of the new methods. The results of tensile and compressive moduli from the two new methods were compared with the direct tension moduli tests and unconfined compression moduli tests. The correlations among them were analyzed. The result indicates that the tensile and compression moduli of asphalt mixtures show a significant difference, and can be obtained by two methods simultaneously. The two new methods realize the synchronous measuring of various moduli of asphalt mixtures.
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An insightful mechanics-based concept is developed for probing the frequency-dependence in in-plane elastic moduli of microstructured lattice materials. Closed-form expressions for the complex ...elastic moduli are derived as a function of frequency by employing the dynamic stiffness matrix of beam elements, which can exactly capture the sub-wavelength scale dynamics. It is observed that the two Poisson's ratios are not dependent on the frequency of vibration, while the amplitude of two Young's moduli and shear modulus increase significantly with the increase of frequency. The variation of frequency-dependent phase of the complex elastic moduli is studied in terms of damping factors of the intrinsic material. The tunable frequency-dependent behaviour of elastic moduli in lattice materials could be exploited in the pseudo-static design of advanced engineering structures which are often operated in a vibrating environment. The generic concepts presented in this paper introduce new exploitable dimensions in the research of engineered materials for potential applications in various vibrating devices and structures across different length-scales.
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Mechanical properties, alpha particles, gamma-ray, proton, and neutron interaction parameters of 40TeO2-(60-x)V2O5-xMoO3: 20 ≤ xMoO3 ≤ 60 mol% (TVM20-TVM60) semiconductor glasses have been ...investigated. Based on Makishima–Mackenzie's model, the total ionic packing density (Vt) and the total dissociation energy (Gt) for TVM-glasses have been computed. Elastic moduli, hardness, and Poisson's ratio haven been calculated. Utilizing WinXcom and EXABCal computer codes, mass attenuation coefficient (MAC), linear attenuation coefficient (LAC), half value layer (HVL), mean free path (MFP), effective atomic number (Zeff), equivalent atomic number (Zeq), energy absorption and exposure built up factors (EABF and EBF), and fast neutron removal cross section ∑R have been computed. Results reflected that the (Vt) of the TVM-glasses varied from 0.597 to 0.610 (m3/mol), while the (Gt) increased from 63.36 × 106 to 63.48 × 106 (KJ/m3) for TVM20 to TVM60 glasses. The highest elastic features were found for TVM60 glass sample with highest value of MoO3 content. The elastic properties varied from 75.77 to 77.45 GPa for Young's modulus, from 54.67 to 56.70 GPa for bulk modulus, and from 0.267 to 0.272 for Poisson's ratio. The TVM60 glass sample possess the highest MAC, followed by TVM50, TVM40, TVM30, and TVM20, respectively. The maximum HVL was obtained at 8 MeV for all glass samples with values of 5.75, 5.30, 5.05, 4.71 and 4.31 cm for TVM 20, TVM30, TVM40, TVM50, and TVM60, respectively. The TVM60 has a better fast neutron shielding capacity compare to the other glasses. The relative difference between EABF and EBF of the glasses were in the order TVM20 > TVM30 > TVM40 > TVM50 > TVM60. We can say that TVM60 glass can attenuate more photons than TVM20-TVM50 glasses.
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A generic analytical framework is proposed to obtain the dynamic elastic moduli of lattice materials under steady-state vibration conditions. The dynamic deformation behaviour of the individual beam ...elements of a lattice is distinct from the behaviour under a static condition. This leads to a completely different global deformation pattern of the lattice material and subsequently opens up a tremendous opportunity to modulate amplitude and phase of the elastic properties of lattices as a function of the ambient vibration. The dynamic stiffness approach proposed in this article precisely captures the sub-wavelength scale dynamics of the periodic network of beams in a lattice material using a single beam-like member. Here the dynamic stiffness matrix of a damped beam element based on the Timoshenko beam theory along with axial stretching is coupled with the unit cell-based approach to derive the most general closed-form analytical formulae for the elastic moduli of lattice materials across the whole frequency range. It is systematically shown how the general expressions of dynamic elastic moduli can be reduced to different special cases by neglecting axial and shear deformations under dynamic as well as classical static conditions. The significance of developing the dynamic stiffness approach compared to conventional dynamic finite element approach is highlighted by presenting detailed analytical derivations and representative numerical results. Further, it is shown how the analytical framework can be readily extended to lattices with non-prismatic beam elements with any spatial variation in geometry and intrinsic material properties. In general, research activities in the field of lattice metamaterials dealing with elastic properties revolve around intuitively designing the microstructural geometry of the lattice structure. Here we propose to couple the physics of deformation as a function of vibrating frequency along with the conventional approach of designing microstructural geometry to expand the effective design space significantly. The stretching-enriched physics of deformation in the lattice materials in addition to the bending and shear deformations under dynamic conditions lead to complex-valued elastic moduli due to the presence of damping in the constituent material. The amplitude, as well as the phase of effective elastic properties of lattice materials, can be quantified using the proposed approach. The dependence of Poisson's ratio on the intrinsic material physics in case of a geometrically regular lattice is found to be in contrary to the common notion that Poisson's ratios of perfectly periodic lattices are only the function of microstructural geometry. The generic analytical approach for analysing the elastic moduli is applicable to any form of two- or three-dimensional lattices, and any profile of the constituent beam-like elements (different cross-sections as well as spatially varying geometry and intrinsic material properties) through a wide range of frequency band. The closed-form expressions of elastic moduli derived in this article can be viewed as the broadband dynamic generalisation of the well-established classical expressions of elastic moduli under static loading, essentially adding a new exploitable dimension in the metamaterials research in terms of dynamics of the intrinsic material.
•A generic analytical framework is proposed for the dynamic elastic moduli of lattice materials under steady-state vibration conditions. The closed-form expressions of elastic moduli derived in this article are to be viewed as the broadband dynamic generalization of the classical expressions of elastic moduli under static loading.•The stretching-enriched physics of deformation in the lattice materials in addition to the bending and shear deformations under dynamic conditions lead to complex elastic moduli, which can be accurately captured using the proposed framework in a broad band of frequency.•The analytical framework reported here is the most general to date; it is applicable to (a) any form of two or three dimensional lattices, (b) any profile of the constituent beam-like elements (different cross sections as well as spatially varying geometry and intrinsic material properties), (c) a wide range of frequency band covering low (including zero) to high frequencies, and (d) includes the effect of intrinsic material damping.•We propose to couple the physics of dynamic deformation along with the conventional approach of designing microstructural geometry to expand the effective design space of lattice materials by seamlessly incorporating the frequency-dependent behaviour.•This paper reveals a dependence of the Poisson's ratios on the intrinsic material physics in case of geometrically regular lattices, which is in contrary to the common notion that Poisson's ratios of perfectly periodic lattices are only function of the microstructural geometry.
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Motivated by constructing moduli spaces of unstable objects, we use new ideas in non-reductive GIT to construct quotients by parabolic group actions. For moduli problems with semistable moduli spaces ...constructed by reductive GIT, we consider associated instability (or HKKN) stratifications, which are often closely related to Harder-Narasimhan stratifications, and construct quotients of the unstable strata under various stabiliser assumptions by further developing ideas of non-reductive GIT. Our approach is to construct parabolic quotients in stages, in order for the required stabiliser assumptions to be more readily verified. To illustrate these ideas, we construct moduli spaces for certain sheaves of fixed Harder-Narasimhan type on a projective scheme in cases where our stabiliser assumptions can be verified.
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We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is étale-locally a quotient stack in a neighborhood of a point with a linearly ...reductive stabilizer group. The proof uses an equivariant version of Artin's algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.
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