Euler Form Research Articles

Cosmology with scalar–Euler form coupling

A coupling between the spacetime geometry and a scalar field involving the Euler 4-form can have important consequences in General Relativity. The coupling is a four-dimensional version of the Jackiw–Teitelboim action, in which a scalar couples to the Euler 2-form in two dimensions. In this case, the first-order formalism, in which the vierbein (or the metric) and the spin connection (or the affine connection) are varied independently, is not equivalent to the second-order one, where the geometry is completely determined by the metric. This is because the torsion postulate Ta ≡ 0 is not valid now and one cannot algebraically solve the spin connection from its own field equation. The direct consequence of this obstruction is that the torsion becomes a new source for the metric curvature, and even if the scalar field is varying very slowly over cosmic scales so as to have no observable astronomical effects at the galactic scale, it has important dynamical effects that can give rise to a cosmological evolution radically different from the standard Friedmann–Robertson–Walker–Lemaitre model.

Asymptotic behavior of neutral delay differential equation of euler form with constant impulsive jumps

In this paper, we investigate the asymptotic behavior of solutions for a class of forced nonlinear neutral differential equation in first-order Euler form with constant impulsive jumps and unbounded delay. Several new sufficient conditions are given to guarantee that every non-oscillatory/oscillatory solution of system tends to zero as t→∞. Our conclusions improve and generalize some related existing results in the literatures.

On Sha's Secondary Chern–Euler Class

AbstractFor a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern– Euler class and was used by Sha to formulate a relative Poincaré–Hopf theorem under the condition that the metric on the manifold is locally product near the boundary. We show that the secondary Chern–Euler form is exact away from the outward and inward unit normal vectors of the boundary by explicitly constructing a transgression form. Using Stokes’ theorem, this evaluates the boundary term in Sha's relative Poincaré–Hopf theorem in terms of more classical indices of the tangential projection of a vector field. This evaluation in particular shows that Sha's relative Poincaré–Hopf theorem is equivalent to the more classical law of vector fields.

A geometric construction of Coxeter-Dynkin diagrams of bimodal singularities

We consider the Berglund-Hubsch transpose of a bimodal invertible polynomial and construct a triangulated category associated to the compactification of a suitable deformation of the singularity. This is done in such a way that the corresponding Grothendieck group with the (negative) Euler form can be described by a graph which corresponds to the Coxeter-Dynkin diagram with respect to a distinguished basis of vanishing cycles of the bimodal singularity.

Study on Transient Properties of Levitated Object in Near-Field Acoustic Levitation

A new approach to the study on the transient properties of the levitated object in near-field acoustic levitation (NFAL) is presented. In this article, the transient response characteristics, including the levitated height of an object with radius of 24 mm and thickness of 5 mm, the radial velocity and pressure difference of gas at the boundary of clearance between the levitated object and radiating surface (squeeze film), is calculated according to several velocity amplitudes of radiating surface. First, the basic equations in fluid areas on Arbitrary Lagrange—Euler (ALE) form are numerically solved by using streamline upwind petrov galerkin (SUPG) finite elements method. Second, the formed algebraic equations and solid control equations are solved by using synchronous alternating method to gain the transient messages of the levitated object and gas in the squeeze film. Through theoretical and numerical analyses, it is found that there is a oscillation time in the transient process and that the response time does not simply increase with the increasing of velocity amplitudes of radiating surface. More investigations in this paper are helpful for the understanding of the transient properties of levitated object in NFAL, which are in favor of enhancing stabilities and responsiveness of levitated object.

Necessary and sufficient conditions for the oscillation of a first-order neutral differential equation of Euler form

Necessary and sufficient conditions for the oscillation of a first-order neutral differential equation of Euler form

Numerical simulation on foam ceramic blasting block device under the action of explosion transform

The responsive behaviour of the action of under explosion blasting block damaging device structure was investigated based on quality equation, energy conservation equation and thermodynamics equation, using the ANSYS finite element simulation and Euler form of momentum equation. This paper analyzed the destructive situation of the inside and outside of the foam ceramic device. The research shows that the finite element simulation results and the numerical calculation results agree, which provides important reference basis in the anti-explosion design and improvement of cut off the device for blasting.

The Implicit Euler Form of Walker Viscoplastic Constitutive Model

In this paper, in the Walker viscoplastic constitutive model equations are rewritten and the implicit Euler integration form of the Walker model is obtained. Furthermore, using this integration form, the subroutine HYPELA in MSC.Marc software has been coded. The finite element analyses, using subroutine HYPELA have been performed on HASTELLOY-X material, and the comparison between the implicit Euler integration and the explicit Euler integration has been done.

Asymptotic behavior of solutions of a first-order impulsive neutral differential equation in Euler form

This paper is concerned with an impulsive neutral differential equation of Euler form with unbounded delays (∗) { d d t [ x ( t ) − C ( t ) x ( α t ) ] + P ( t ) t x ( β t ) = 0 , t ≥ t 0 > 0 , t ≠ t k , x ( t k ) = b k x ( t k − ) + ( 1 − b k ) ∫ β t k t k P ( s / β ) s x ( s ) d s , k = 1 , 2 , … . Sufficient conditions are obtained for every solution of (∗) to tend to a constant as t → ∞ .

Euler integrals and multi-integrals of linear partial differential equations

In this paper, we discuss the notion of reducibility of solutions of the Euler form generalizing solutions arising from the Laplace cascade method for integrating second-order hyperbolic equations in the plane. We present reduction algorithms and prove the equivalence of various possible exact definitions of the reduction of similar explicit solutions.

Sortable elements in infinite Coxeter groups

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group W. We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection \pidown^c mapping each element of W to the unique maximal c-sortable element below it in the weak order. The fibers of \pidown^c essentially define the c-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of \pidown^c. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.

Integral bilinear forms, Coxeter transformations and Coxeter polynomials of finite posets

Linear algebra technique in the study of linear representations of finite posets is developed in the paper. A concept of a quadratic wandering on a class of posets I is introduced and finite posets I are studied by means of the four integral bilinear forms b ˆ I , b I , b ¯ I , b I • : Z I × Z I → Z (1.1), the associated Coxeter transformations, and the Coxeter polynomials (in connection with bilinear forms of Dynkin diagrams, extended Dynkin diagrams and irreducible root systems are also studied). Bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of Dynkin diagrams and extended Dynkin diagrams are discussed. A homological interpretation of the bilinear forms (1.1) is given and Z -bilinear equivalences between them are discussed. By applying well-known results of Bongartz, Loupias, and Zavadskij-Shkabara, we give several characterisations of posets I, with the Euler form q ¯ I ( x ) = b ¯ I ( x , x ) weakly positive (resp. with the reduced Euler form q I • ( x ) = b I • ( x , x ) weakly positive), and posets I, with the Tits form q ˆ I ( x ) = b ¯ I ( x , x ) weakly positive.

Oscillation criteria for a first-order impulsive neutral differential equation of Euler form

In this paper, we are concerned with the oscillation of a first-order impulsive neutral differential equation of Euler form with variable delays. Our results reveal the fact that the oscillatory behavior of all solutions of differential equations without impulses can be inherited by impulsive differential equations under certain impulsive perturbations. It is also seen that the oscillatory properties of all solutions of impulsive differential equations may be caused by the impulsive perturbations, though the corresponding differential equations without impulses admit a nonoscillatory solution. Some examples are also given to illustrate the applicability of the results obtained.

Lp estimates for quantities advected by a compressible flow

We consider the evolution of a quantity advected by a compressible flow and subject to diffusion. When this quantity is scalar it can be, for instance, the temperature of the flow or the concentration of some pollutants. Because of the diffusion term, one expects the equations to have a regularizing effect. However, in their Euler form, the equations describe the evolution of the quantity multiplied by the density of the flow. The parabolic structure is thus degenerate near vacuum (when the density vanishes). In this paper we show that we can nevertheless derive uniform L p bounds that do not depend on the density (in particular the bounds do not degenerate near vacuum). Furthermore the result holds even when the density is only a measure. We investigate both the scalar and the system case. In the former case, we obtain L ∞ bounds. In the latter case the quantity being investigated could be the velocity field in compressible Navier–Stokes type of equations, and we derive uniform L p bounds for some p depending on the ratio between the two viscosity coefficients (the main additional difficulty in that case being to deal with the second viscosity term involving the divergence of the velocity). Such estimates are, to our knowledge, new and interesting since they are uniform with respect to the density. The proof relies mostly on a method introduced by De Giorgi to obtain regularity results for elliptic equations with discontinuous diffusion coefficients.

Oscillation of First-Order Neutral Differential Equations With Unbounded Delay and Euler Form

Oscillation of First-Order Neutral Differential Equations With Unbounded Delay and Euler Form

Nonlinear ferro-electro-elastic beam theory

Motivated by the uniqueness and potential of the nonlinear range of piezoelectric and ferroelectric smart materials and structures, a static physically nonlinear ferro-electro-elastic beam theory which takes the effect of domain switching into account is developed. The kinematic assumptions adopt the geometrically linear Bernoulli–Euler form for the mechanical components and a first-order theory for the electrical potential, and lay the basis for further augmentation to higher order theories. The beam theory includes the field equations that correspond to the static case, the boundary conditions and the constitutive equations of ferro-electro-elasticity. The general 3-D constitutive equations are reduced to comply with the beam theory and formulated as ordinary differential equations by means of a set of generalized electro-mechanical stiffnesses. A micromechanical constitutive model that accounts for the loading history and for the domain switching phenomenon is adopted and an iterative solution procedure that incorporates the micromechanical approach is suggested. A numerical example that demonstrates the impact of the domain switching on the nonlinear electromechanical static response of a ferro-electro-elastic beam is presented and discussed. The quantitative assessment of this behavior takes a step towards new structural applications that cope with or even take advantage of the nonlinear ferro-electro-elastic range.

Notes on 2-parameter quantum groups I

A simpler definition for a class of 2-parameter quantum groups associated to semisimple Lie algebras is given in terms of Euler form. Their positive parts turn out to be 2-cocycle deformations of each other under some conditions. An operator realization of the positive part is given.

On the periodicity of Coxeter transformations and the non-negativity of their Euler forms

We show that for piecewise hereditary algebras, the periodicity of the Coxeter transformation implies the non-negativity of the Euler form. Contrary to previous assumptions, the condition of piecewise heredity cannot be omitted, even for triangular algebras, as demonstrated by incidence algebras of posets. We also give a simple, direct proof, that certain products of reflections, defined for any square matrix A with 2 on its main diagonal, and in particular the Coxeter transformation corresponding to a generalized Cartan matrix, can be expressed as - A + - 1 A - t , where A + , A - are closely associated with the upper and lower triangular parts of A.

Integrals of equivariant forms and a Gauss–Bonnet theorem for constructible sheaves

The Berline–Vergne integral localization formula for equivariantly closed forms ([N. Berline, M. Vergne, Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295 (1982) 539–541], Theorem 7.11 in [N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag, 1992]) is well-known and requires the acting Lie group to be compact. In this article, we extend this result to real reductive Lie groups G R . As an application of this generalization, we prove an analogue of the Gauss–Bonnet theorem for constructible sheaves. If F is a G R -equivariant sheaf on a complex projective manifold M , then the Euler characteristic of M with respect to F χ ( M , F ) = 1 ( 2 π ) dim C M ∫ Ch ( F ) χ g C ˜ as distributions on g R , where Ch ( F ) is the characteristic cycle of F and χ g C ˜ is the Euler form of M extended to the cotangent space T ∗ M (independently of F ). We also consider an analogue of Duistermaat–Heckman measures for real reductive Lie groups acting on symplectic manifolds. In [M. Libine, Riemann–Roch–Hirzebruch integral formula for characters of reductive Lie groups, Represent. Theory 9 (2005) 507–524. Also math.RT/0312454] I apply the results of this article to obtain a Riemann–Roch–Hirzebruch type integral formula for characters of representations of reductive groups.

Oscillation of first order neutral differential equations of Euler form

In this paper, we investigate the first order neutral differential equation of Euler form with variable unbounded delaywhere 0 ≤