Let h2 = Sp(4,R)/U(2) be the Siegel upperhalf space of rank 2. The quotient Sp(4,Z)\h2 has three remarkable properties: (a) it is the moduli space of principally polarized abelian surfaces, (b) it has the structure of a quasi-projective complex algebraic variety which is defined over the rational numbers Q, and (c) it has a natural compactification (the BailyBorel Satake compactification) which...

In this paper, we first define an equivalence relation on the sequence space $\Sigma _{2}$. Then equip quotient set _{2}/_{\sim}$ with a metric $d_1$. We also determine isometry map between spaces $(\Sigma _{2}/_{\sim},d_1)$ and $([0,1],d_{eucl})$. Finally, investigate symmetry conditions respect to some points compare truncation errors for computations which is obtained by metrics $d_{eucl}$

We study the existence of symplectic resolutions of quotient singularities V/G, where V is a symplectic vector space and G acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the form KoS2 where K < SL2(C), for which the corresponding quotient singularity admits a projective symplectic resolution. As a consequence, for dimV 6= 4, we...

New concepts related to approximating a Lipschitz function between Banach spaces by a ne functions are introduced Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X Y for which any Lipschitz function from X to Y can be so approx imated is obtained This is applied to the study of Lipschitz and uniform quotient mapp...

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient ...

The essential matrix, which encodes the epipolar constraint between points in two projective views, is a cornerstone of modern computer vision. Previous works have proposed different characterizations of the space of essential matrices as a Riemannian manifold. However, they either do not consider the symmetric role played by the two views or do not fully take into account the geometric peculia...

This short, expository note proves existence of the maximal quotient of a variety by free rational curves. 1. Definition of a maximal free rational quotient Definition 1.1. Let V be a Deligne-Mumford stack over a field k, and denote the smooth locus by V sm ⊂ V . A 1-morphism f : Pk → V sm is a free rational curve to V if fTV is generated by global sections and has positive degree. Let S be an ...

in this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  we relate the cohomology of a normal subgroup $n$ of a topological group $g$ and the quotient $g/n$ to the cohomology of $g$. we get the inflation-restriction exact sequence. also, we obtain a seven-term exact cohomology sequence up to dimension 2. we give an interpretation of the first non-a...

In this paper a new notion of topological spaces namely, I-sequential topological spaces is introduced and investigated. This new space is a strictly weaker notion than the first countable space. Also I-sequential topological space is a quotient of a metric space.

Every uniformly exhaustive submeasure is equivalent to a measure. From this, we deduce that every vector measure with compact range in an F-space has a control measure. We also show that co (or any E.-space) is a T;space, i.e. cannot be realized as the quotient of a nonlocally convex F-space by a one-dimensional subspace.