Auswahl der wissenschaftlichen Literatur zum Thema „GV invariant“

If the vorticity field of an ideal fluid is tangent to a foliation, additional conservation laws arise. For a class of zero-helicity vorticity fields, the Godbillon-Vey (GV) invariant of foliations is defined and is shown to be an invariant purely of the vorticity, becoming a higher-order helicity-type invariant of the flow. GV ≠ 0 gives both a global topological obstruction to steady flow and, in a particular form, a local obstruction. GV is interpreted as helical compression and stretching of vortex lines. Examples are given where the value of GV is determined by a set of distinguished closed vortex lines.

An action α of a discrete group Γ on the circle S1 as orientation preserving C∞-diffeomorphisms gives rise to a foliation on the homotopy quotient S1Γ, and its Godbillon-Vey invariant is, by definition, a cohomology class of S1Γ([1]). This cohomology class naturally defines an additive map from the geometric K-group K0(S1, Γ) into C, through the Chern character from K0(S1, Γ) to H*(S1, Γ Q).Using cyclic cohomology, Connes constructed in [2] an additive map, GV(α), which we shall call the Godbillon-Vey map, from the K0-group of the reduced crossed product C*-algebra C(S1) ⋊ αΓ into C. He showed that GV(α) agrees with the geometric Godbillon-Vey invariant through the index map μ from K0(S1, Γ) to K0(C(S1) ⋊ αΓ).

A maximum electric field E = 2.65 GV/m with an accelerated electron current of 1 KA has been obtained, for pulse lengths of 130 ps, in an electron gun based on Pulse Power Technology. This is the highest accelerating field ever achieved in the presence of such a large current. Measurements of beam emittance and energy from 0.4 to 2.65 MeV show that the scaling of the invariant emittance with electric field and with beam current is consistent with theoretical predictions. A few applications of high-gradient acceleration are discussed.

Abstract In view of the importance of tropical rainfall and the ubiquitous need for its estimates in climate modeling, the authors assess the ability of the Tropical Rainfall Measuring Mission (TRMM) precipitation radar (PR) to characterize the scaling characteristics of rainfall by comparing the derived results with those obtained from the ground-based radar (GR) data. The analysis is based on 59 months of PR and GR rain rates at three TRMM ground validation (GV) sites: Houston, Texas; Melbourne, Florida; and Kwajalein Atoll, Republic of the Marshall Islands. The authors consider spatial scales ranging from about 4 to 64 km at a fixed temporal scale corresponding to the sensor “instantaneous” snapshots (∼15 min). The focus is on the scaling of the marginal moments, which allows estimation of the scaling parameters from a single scene of data. The standard rainfall products of the PR and the GR are compared in terms of distributions of the scaling parameter estimates, the connection between the scaling parameters and the large-scale spatial average rain rate, and deviations from scale invariance. The five main results are as follows: 1) the PR yields values of the rain intermittence scaling parameter within 20% of the GR estimate; 2) both the PR and GR data show a one-to-one relationship between the intermittence scaling parameter and the large-scale spatial average rain rate that can be fit with the same functional form; 3) the PR underestimates the curvature of the scaling function from 20% to 50%, implying that high rain-rate extremes would be missed in a downscaling procedure; 4) the majority of the scenes (>85%) from both the PR and GR are scale invariant at the moment orders q = 0 and 2; and 5) the scale-invariance property tends to break down more likely over ocean than over land; the rainfall regimes that are not scale invariant are dominated by light storms covering large areas. Our results further show that for a sampling size of one year of data, the TRMM temporal sampling does not significantly affect the derived scaling characteristics. The authors conclude that the TRMM PR has the ability to characterize the basic scaling properties of rainfall, though the resulting parameters are subject to some degree of uncertainty.

In analogy with the Gopakumar–Vafa (GV) conjecture on Calabi–Yau (CY) 3-folds, Klemm and Pandharipande defined GV type invariants on Calabi–Yau 4-folds using Gromov–Witten theory and conjectured their integrality. In a joint work with Maulik and Toda, the author conjectured their genus zero invariants are [Formula: see text] invariants of one-dimensional stable sheaves. In this paper, we study this conjecture on the total space of canonical bundle of a Fano 3-fold [Formula: see text], which reduces to a relation between twisted GW and [Formula: see text] invariants on [Formula: see text]. Examples are computed for both compact and non-compact Fano 3-folds to support our conjecture.

We have systematically investigated a class of models which is characterised by Euler-Lagrange equations for the quark fields (Dirac equation) which contain bag-like (i.e. Lorentz-scalar) confining potentials and various Lorentz-vector (Coulomb-like and modified linear) confining potentials and report the results for gA/gV, [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and ρ(r) (quark annihilation density). It is demonstrated that the effects of the Lorentz-vector potential on low-energy observables are naturally limited in the baryon sector by the "Klein paradox". A strong Lorentz-scalar confining potential can, however, tolerate a small admixture of a Lorentz-vector component. The one-gluon-exchange Lorentz-vector potential is characterized by two parameters αs and r0. The scale-invariance of gA/gV is manifest as a peculiar r0–independence. With Lorentz-vector potentials of the Coulomb form, gA/gV is independent of the quark mass. gA and the quark eigen-energy E0 (and hence also the nucleon mass) decrease considerably for increasing positive αs.

The Godbillon–Vey invariant occurs in homology theory, and algebraic topology, when conditions for a co-dimension 1, foliation of a three-dimensional manifold are satisfied. The magnetic Godbillon–Vey helicity invariant in magnetohydrodynamics (MHD) is a higher-order helicity invariant that occurs for flows in which the magnetic helicity density $h_{m}=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{B}=\boldsymbol{A}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \boldsymbol{A})=0$ , where $\boldsymbol{A}$ is the magnetic vector potential and $\boldsymbol{B}$ is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon–Vey field $\unicode[STIX]{x1D6C8}=\boldsymbol{A}\times \boldsymbol{B}/|\boldsymbol{A}|^{2}$ and the Godbillon–Vey helicity density $h_{\text{gv}}=\unicode[STIX]{x1D6C8}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\times \unicode[STIX]{x1D6C8})$ in general MHD flows in which either $h_{m}=0$ or $h_{m}\neq 0$ . A conservation law for $h_{\text{gv}}$ occurs in flows for which $h_{m}=0$ . For $h_{m}\neq 0$ the evolution equation for $h_{\text{gv}}$ contains a source term in which $h_{m}$ is coupled to $h_{\text{gv}}$ via the shear tensor of the background flow. The transport equation for $h_{\text{gv}}$ also depends on the electric field potential $\unicode[STIX]{x1D713}$ , which is related to the gauge for $\boldsymbol{A}$ , which takes its simplest form for the advected $\boldsymbol{A}$ gauge in which $\unicode[STIX]{x1D713}=\boldsymbol{A}\boldsymbol{\cdot }\boldsymbol{u}$ where $\boldsymbol{u}$ is the fluid velocity. An application of the Godbillon–Vey magnetic helicity to nonlinear force-free magnetic fields used in solar physics is investigated. The possible uses of the Godbillon–Vey helicity in zero helicity flows in ideal fluid mechanics, and in zero helicity Lagrangian kinematics of three-dimensional advection, are discussed.

Abstract We propose a new way to compute the genus zero Gopakumar-Vafa invariants for two families of non-toric non-compact Calabi-Yau threefolds that admit simple flops: Reid’s Pagodas, and Laufer’s examples. We exploit the duality between M-theory on these threefolds, and IIA string theory with D6-branes and O6-planes. From this perspective, the GV invariants are detected as five-dimensional open string zero modes. We propose a definition for genus zero GV invariants for threefolds that do not admit small crepant resolutions. We find that in most cases, non-geometric T-brane data is required in order to fully specify the invariants.

Abstract We present a novel technique to obtain base independent expressions for the matter loci of fibrations of complete intersection Calabi-Yau onefolds in toric ambient spaces. These can be used to systematically construct elliptically and genus one fibered Calabi-Yau d-folds that lead to desired gauge groups and spectra in F-theory. The technique, which we refer to as GV-spectroscopy, is based on the calculation of fiber Gopakumar-Vafa invariants using the Batyrev-Borisov construction of mirror pairs and application of the so-called Frobenius method to the data of a parametrized auxiliary polytope. In particular for fibers that generically lead to multiple sections, only multi-sections or that are complete intersections in higher codimension, our technique is vastly more efficient than classical approaches. As an application we study two Higgs chains of six-dimensional supergravities that are engineered by fibrations of codimension two complete intersection fibers. Both chains end on a vacuum with G = ℤ4 that is engineered by fibrations of bi-quadrics in ℙ3. We use the detailed knowledge of the structure of the reducible fibers that we obtain from GV-spectroscopy to comment on the corresponding Tate-Shafarevich group. We also show that for all fibers the six-dimensional supergravity anomalies including the discrete anomalies generically cancel.

Abstract The conifold is a basic example of a noncompact Calabi-Yau threefold that admits a simple flop, and in M-theory, gives rise to a 5d hypermultiplet at low energies, realized by an M2-brane wrapped on the vanishing sphere. We develop a novel gauge-theoretic method to construct new classes of examples that generalize the simple flop to so-called length ℓ = 1, . . . , 6. The method allows us to naturally read off the Gopakumar-Vafa invariants. Although they share similar properties to the beloved conifold, these threefolds are expected to admit M2-bound states of higher degree ℓ. We demonstrate this through our computations of the GV invariants. Furthermore we characterize the associated Higgs branches by computing their dimensions and flavor groups. With our techniques we extract more refined data such as the charges of the hypers under the flavor group.