Intersection Index Research Articles

The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties

For an irreducible subvariety Z in an algebraic group G we define a nonnegative integer gdeg(Z) as the degree, in a certain sense, of the Gauss map of Z. It can be regarded as a substitution for the intersection index of the conormal bundle to Z with the zero section of T^*G, even though G may be non-compact. For G a semiabelian variety (in particular, an algebraic torus (C^*)^n) we prove a Riemann-Roch-type formula for constructible sheaves on G, which involves our substitutions for the intersection indices. As a corollary, we get that a perverse sheaf on such a G has nonnegative Euler characteristic, generalizing a theorem of Loeser-Sabbah.

2.5 kDa and 5.0 kDa heparin fragments specifically inhibit microvessel sprouting and network formation in VEGF165-mediated mammalian angiogenesis.

Tumour growth is angiogenesis dependent. Thrombosis and thromboembolism are very common in cancer patients. These patients are often treated with heparin as an anti-coagulant. Many tumour angiogens, including VEGF165, and endogenous anti-angiogenesis factors bind heparin tightly. Using the non-surgical mesenteric-window angiogenesis assay, we studied in detail the systemic effect of heparin fractions with a mean MW of 2.5, 5.0 and 16.4 kDa on the microvessel sprouting and network formation in angiogenesis mediated by VEGF165 in rats. The microvessel network was assessed objectively in terms of the number and lengths of segments (the distance between two successive branching points), the number of branching points, the degree of tortuosity, the index of interconnecting loop formation, the index of intersection, as well as the number and lengths of sprouts. Compared with the saline control, the 2.5 kDa fraction significantly shortened the microvessel sprouts and the microvessel segments but increased the microvessel tortuosity in statistical terms; the 5.0 kDa fraction statistically significantly shortened the sprouts, decreased the number of segments and the number of microvessel branching points; whereas the 16.4 kDa fraction statistically significantly elongated the longest segments. Moreover, statistically significant differences were found between the three heparin fractions in terms of microvessel tortuosity (2.5 vs. 16.4 kDa), index of loop formation (5.0 vs. 2.5 + 16.4 kDa) and index of intersection (5.0 vs. 16.4 kDa). These findings demonstrate that heparin fragments size-specifically inhibit microvessel sprouting and network formation in VEGF165-mediated angiogenesis. As VEGF165 is a potent angiogen in human tumours, we suggest that heparin enriched in 2.5 kDa species and 5.0 kDa species especially should be exploited as a combined anti-coagulant and specific adjuvant anti-angiogenic agent in cancer patients who require anti-coagulant therapy.

NO and de novo mammalian angiogenesis: further evidence that NO inhibits bFGF-induced angiogenesis while not influencing VEGF165-induced angiogenesis.

Using the non-surgical rat mesenteric window angiogenesis assay, we studied the systemic effect of (i) the nitric oxide (NO)-releasing vasodilator isosorbide-5-mononitrate (ISMN) and (ii) the NO-synthase inhibitor L-NAME on angiogenesis induced by the intraperitoneal injection of bFGF and VEGF165. The response was assessed objectively and quantitatively by microscopic morphometry and image analysis in terms of the vascularized area (VA; a measurement of microvessel spatial extension), the microvascular length (MVL; a composite measurement of microvessel density), the total microvascular length (TMVL=VA x MVL), the number of microvessel segments per unit tissue volume (No. MS), the length of the microvessel segments (Le. MS) and the degree of microvessel tortuosity (MVT). Additional architectural features of the network were assessed in terms of variables introduced here: the number of microvessel branching points per unit tissue volume (No. BP), the index of interconnecting microvessel loop formation (In. LF), the index of microvessel intersection (In. IS), the number of microvessel sprouts per unit tissue volume (No. SP) and their length (Le. SP). In bFGF-mediated angiogenesis, L-NAME significantly, augmented angiogenesis, whereas ISMN significantly inhibited angiogenesis. By contrast, neither L-NAME nor ISMN affected the angiogenic response to VEGF165.

On the intersection product of analytic cycles

We prove that the generalized index of intersection of an analytic set with a closed submanifold (Thm. 4.3) and the intersection product of analytic cycles (Thm. 5.4), which are defined in [T₂], are intrinsic. We define the intersection product of analyti

Linking Numbers and Boundaries of Varieties

called a cornerstone of algebraic geometry ([GH, p. 62]), is a simple consequence of the fact that analytic varieties carry a natural orientation. Recast in terms of linking numbers, it is our principal motivation. It implies the following: Let M be a smooth oriented compact 3-manifold in C3. Suppose that M a bounded complex 2-variety V. Here bounds means, in the sense of Stokes' theorem, i.e., that b[V] = [M] as currents. Let A be an algebraic curve in C3 which is disjoint from M. Consider the linking number link(M, A) of M and A. Since this linking number is equal to the intersection number (i.e. the sum of the intersection indices) of V and A, by the positivity

On the WDVV equation in quantum K-theory.

0. Introduction. Quantum cohomology theory can be described in general words as intersection theory in spaces of holomorphic curves in a given Kahler or almost Kahler manifold X. By quantum K-theory we may similarly understand a study of complex vector bundles over the spaces of holomorphic curves in X. In these notes, we will introduce a K-theoretic version of the Witten-Dijkgraaf-Verlinde-Verlinde equation which expresses the associativity constraint of the “quantum multiplication” operation on K(X). Intersection indices of cohomology theory, ∫

String holonomy and extrinsic geometry in four-dimensional topological gauge theory

The most general gauge-invariant marginal deformation of four-dimensional abelian BF-type topological field theory is studied. It is shown that the deformed quantum field theory is topological and that its observables compute, in addition to the usual linking numbers, smooth intersection indices of immersed surfaces which are related to the Euler and Chern characteristic classes of their normal bundles in the underlying space-time manifold. Canonical quantization of the theory coupled to non-dynamical particle and string sources is carried out in the Hamiltonian formalism and explicit solutions of the Schr̈odinger equation are obtained. The wavefunctions carry a one-dimensional unitary representation of the particle-string exchange holonomies and of non-topological string-string intersection holonomies given by adiabatic limits of the world-sheet Euler numbers. They also carry a multi-dimensional projective representation of the deRham complex of the underlying spatial manifold and define a generalization of the presentation of its motion group from Euclidean space to an arbitrary 3-manifold. Some potential physical applications of the topological field theory as a dual model for effective vortex strings are discussed.

Relations Between the Correlators of the Topological Sigma-Model Coupled to Gravity

We prove a new recursive relation between the correlators ¶ , which together with known relations allows one to express all of them through the full system of Gromov–Witten invariants in the sense of Kontsevich–Manin and the intersection indices of tautological classes on , effectively calculable in view of earlier results due to Mumford, Kontsevich, Getzler, and Faber. This relation shows that a linear change of coordinates of the big phase space transforms the potential with gravitational descendants to another function defined completely in terms of the Gromov–Witten correspondence and the intersection theory on . We then extend the formalism of gravitational descendants from quantum cohomology to more general Frobenius manifolds and Cohomological Field Theories.

Matrix Model Tools and Geometry of Moduli Spaces

The relation between matrix models and geometrical structures on moduli spaces of algebraic curves is reviewed. The description of discretized moduli spaces and the generating function for intersection indices (cohomological classes) on these spaces is presented by the matrix model with the logarithmic potential when using the stratification procedure to express integrals over open strata via integrals over compactified spaces. We review the constraint equations and related Virasoro algebras appearing in various matrix models paying the special attention to the interrelations between these models. The explicit solutions to matrix models in 1/N expansion are presented.

Matrix model for discretized moduli space

We study the algebraic geometrical background of the Penner-Kontsevich matrix model with the potential Nα tr[− 1 2 ΛXΛX+log(1− X+ X]. We show that this model describes intersection indices of linear bundles on the discretized moduli space just in the same fashion as the Kontsevich model is related to intersection indices (cohomological classes) on Riemann surfaces of arbitrary genus. The special role of the logarithmic potential originating from the Penner matrix model is demonstrated. The boundary effects, which were not essential in the case of the Kontsevich model, are now relevant, and the intersection indices on the discretized moduli space of genus g are expressed through Kontsevich's indices of genus g and of the lower genera using a stratification procedure.

Computation of intersection index of totally real and complex disk with common boundary in ?2 and ?1�?1

The main result of the paper is a classification theorem for two-dimensional spheres obtained from a pair of disks, a totally real and a complex one, glued together.

REALIZATION OF QUADRATIC FORMS BY SMOOTH MANIFOLDS

It is proved in this paper that, for every 1$ SRC=http://ej.iop.org/images/0025-5734/62/1/A12/tex_sm_3234_img1.gif/>, each integral unimodular quadratic form is the intersection index form of some smooth closed manifold of dimension . The question is also studied of the realizability of such forms by manifolds with additional structures on the stable normal bundle and, as a consequence, of the realizability of forms by highly connected manifolds.Bibliography: 10 titles.

Statistical Methods for Evaluating the Geometrical Hierarchy of a Network

The objective of this paper is to present statistical methods for evaluating the geometrical hierarchy of a network in comparison with a random hierarchy network. First, the random hierarchy network is explicitly formulated by four stochastic processes in which line segments or lines, which represent roads, are randomly placed and their placement is independent of their ranks. Second, under these stochastic processes, this paper derives the expected value and variance of i) the number of intersections between roads of different ranks; and ii) the distance from an arbitrary point to the nearest highest‐ranked road through two kinds of routes. Using these expected values, the intersection index and the detour index are proposed, and it is shown that these indicators are useful for evaluating the geometrical hierarchy of a network. As calculation of these indices is laborious, an efficient computational method is developed, and its application to an actual example is described.

ON THE INTERSECTION INDEX OF DIVISORS

Using an idea of A. N. Parsin, a formula for the intersection index of Cartier divisors on an arbitrary complete variety is introduced. This formula is analogous to one for the degree of a divisor on a complete nonsingular curve expressed with the aid of canonical discrete valuations. Bibliography: 4 titles.

Homotopy classification of some four-dimensional manifolds

In this paper there is proved a generalization of the results of Whitehead and Pontryagin on the homotopy classification of closed, simply connected four-manifolds. Let W and M be compact four-dimensional simply connected oriented four-manifolds. By qw is denoted the intersection index on the group H2(W).Basic Result. THEOREM (Extension). Let the groups H1(δW)and H1(δM) be finite and suppose given a homotopy equivalence f∶δW→δM. In order that f can be extended to a homotopy equivalence (W,δW)→(M,δM), it is necessary and sufficient that there should exist an isomorphism Ξ, such that the diagram is commutative and Ξ*qm=qm.