Relative Quasimaps and Mirror Formulae

We construct and study the theory of relative quasimaps in genus zero, in the spirit of Gathmann. When $X$ is a smooth toric variety and $Y$ is a smooth very ample hypersurface in $X$, we produce a virtual class on the moduli space of relative quasimaps to $(X,Y)$, which we use to define relative quasimap invariants. We obtain a recursion formula which expresses each relative invariant in terms of invariants of lower tangency, and apply this formula to derive a quantum Lefschetz theorem for quasimaps, expressing the restricted quasimap invariants of $Y$ in terms of those of $X$. Finally, we show that the relative $I$-function of Fan–Tseng–You coincides with a natural generating function for relative quasimap invariants, providing mirror-symmetric motivation for the theory.

Effect of Amount on Expansion Property of MgO-Type Expansive Agent Used in Cement-Based Materials

Previous research indicated that the expansion property of MgO-type expansive agent (MEA) strongly depended on the amount. However, the quantitative effect of amount on expansion property of MEA has not been clearly demonstrated. In the present the expansion value of cement paste with 3%, 4%, 5% and 6% MEA calcinated at 850°C and 1200 °C, and then relative expansion value was introduced to investigated the relationship between the mount and expansion of MEA. The results indicated The relative expansion of MEA calcinated at 850°C was relative invariant to curing temperature and proportional increased with the amount ratio. The relative expansion of MEA calcinated at 1200°C proportional increased with curing age, and the increasing rate rose with the amount ratio of MEA and curing temperature.

INVARIANTS AND COINVARIANTS OF FINITE PSEUDOREFLECTION GROUPS, JACOBIAN DETERMINANTS AND STEENROD OPERATIONS

Let $\rho:G\hookrightarrow\GL(n,\F)$ be a representation of a finite group $G$ over a finite field $\F$ and $f_1,\dots,f_n\in\F[V]^G$ such that the ring of invariants is a polynomial algebra $\F[f_1,\dots,f_n]$. It is known that in this case the algebra of coinvariants $\F[V]_G$ is a Poincaré duality algebra, and if, moreover, the order of $G$ is invertible in $\F$, that a fundamental class is represented by the Jacobian determinant $\mathchoice{\det\biggl[\frac{\partial f_i}{\partial z_j}\biggr]}{\det[\partial f_i/\partial z_j]}{\det[\partial f_i/\partial z_j]} {\det[\partial f_i/\partial z_j]}$, and is therefore a $\det^{-1}$-relative invariant. In this note we deduce what happens in the modular case. As a bonus we obtain a new criterion for an unstable algebra over the Steenrod algebra to be a ring of invariants.AMS 2000 Mathematics subject classification: Primary 13A50; 55S10

Subgroups of products of surface groups

The subgroup structure of a direct product of two Fuchsian groups is very complicated; for example, Baumslag and Roseblade [1] have shown that a direct product Fm1×Fm2 of finitely generated free groups contains continuously many distinct isomorphism classes of finitely generated subgroups. The situation is much simpler, however, if attention is restricted to finitely generated normal subgroups of Fm1×Fm2; then on general grounds one cannot expect to get more than a countable infinity, but, in fact, the situation is almost finite; we showed, in [4], that Fm1×Fm2 contains precisely 1+min{m1, m2} orbits of maximal normal subdirect products under the natural action of its automorphism group.In this paper we study the situation which arises if the free groups Fmi are replaced by fundamental groups of closed surfaces. This question was previously considered by Nigel Carr in the final chapter of his (unpublished) thesis [2]. By appealing to the relative invariant theory of pairs of skew bilinear forms, Carr was able to show that in a direct product Σg1×Σg2 of orientable surface groups of genus [ges ]2 the number of orbits of maximal normal subdirect products is always infinite. Here we refine and extend Carr's approach to study the manner in which the finiteness result of [4] breaks down on passing to more general Fuchsian groups.

Efficient Cues for Discriminating Skewed Curved-Line Segments

Visual discrimination of circular arcs differing in curvature reaches hyperacute levels of performance. What spatial attributes of curved contours provide the necessary visual cue? Statistical-efficiency theory has previously been applied to data on the discrimination of symmetric curved-line segments undergoing expansions and contractions perpendicular to their chords. The results have suggested that relative invariants with respect to these transformations are the best cues, since they accounted for the most variance in the data (a relative invariant is an attribute such that the ratio of its values is constant under transformation). Expansions and contractions are examples of affine transformations, which in general provide a good approximation to the effects of viewpoint change. If some attribute of a curved line is a relative invariant with respect to affine transformations, is it then a good cue? An experiment was performed in which observers discriminated curved-line segments that had been affine transformed by progressive amounts of shear along their chords as well as expansions and contractions along and perpendicular to their chords. Shear can be interpreted as a relative affine invariant, and, since shear destroys symmetry by skewing the curve, it should provide a good cue. In fact, although expansions and contractions proved to be good cues, shear did not. Candidate cues that were not relative affine invariants (eg Euclidean curvature, turning angle) were also poor cues. It appears that being a relative affine invariant is a necessary but not sufficient condition for a cue to be efficient in the discrimination of curved-line segments.

On the convergence of the zeta function for certain prehomogeneous vector spaces

Let (G, V) be an irreducible prehomogeneous vector space defined over a number field k, P ∈ k[V] a relative invariant polynomial, and χ a rational character of G such that . For , let Gx be the stabilizer of x, and the connected component of 1 of Gx. We define L0 to be the set of such that does not have a non-trivial rational character. Then we define the zeta function for (G, Y) by the following integralwhere Φ is a Schwartz-Bruhat function, s is a complex variable, and dg” is an invariant measure.