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We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good minimal model over a Zariski open subset of the base and the relative log canonical model is of maximal variation., 24 pages, Comments welcome

We show a rigidity theorem for the Seiberg-Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of 4-manifolds $M$ for which the inclusion maps $\operatorname{Diff}(M) \hookrightarrow \operatorname{Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series of non-smoothable topological actions on some spin 4-manifolds., 40 pages, to appear in Compositio Mathematica

We classify all supersymmetric solutions of minimal D=4 gauged supergravity with (2,2) signature and a positive cosmological constant which admit exactly one Killing spinor. This classification produces a geometric structure which is more general than that found for previous classifications of N=2 supersymmetric solutions of this theory. We illustrate how the N=2 solutions which consist of a fibration over a 3-dimensional Lorentzian Gauduchon-Tod base space can be written in terms of this more generic geometric structure., 20 pages, latex

We explore how far one can go in constructing $d$-dimensional static black holes coupled to $p$-form and scalar fields before actually specifying the gravity and electrodynamics theory one wants to solve. At the same time, we study to what extent one can enlarge the space of black hole solutions by allowing for horizon geometries more general than spaces of constant curvature. We prove that a generalized Schwarzschild-like ansatz with an arbitrary isotropy-irreducible homogeneous base space (IHS) provides an answer to both questions, up to naturally adapting the gauge fields to the spacetime geometry. In particular, an IHS-K\"ahler base space enables one to construct magnetic and dyonic 2-form solutions in a large class of theories, including non-minimally couplings. We exemplify our results by constructing simple solutions to particular theories such as $R^2$, Gauss-Bonnet and (a sector of) Einstein-Horndeski gravity coupled to certain $p$-form and conformally invariant electrodynamics., Comment: 15 pages. v2: typos fixed, minor improvements to the text, refs. added

For Riemannian submersions, we establish some estimates for the spectrum of the total space in terms of the spectrum of the base space and the geometry of the fibers. In particular, for Riemannian submersions of complete manifolds with closed fibers of bounded mean curvature, we show that the spectrum of the base space is discrete if and only if the spectrum of the total space is discrete., Comment: 18 pages

In this paper we consider the finite groups that act fiber- and orientation-preservingly on closed, compact, and orientable Seifert manifolds that fiber over an orientable base space. We establish a method of constructing such group actions and then show that if an action satisfies a condition on the obstruction class of the Seifert manifold, it can be derived from the given construction. The obstruction condition is refined and the general structure of the finite groups that act via the construction is provided., Comment: 19 pages, 4 figures. Revision after referee's comments

We study regular coverings of graphs and manifolds with a focus on properties of the heat equation. In particular, we look at stochastic incompleteness, the Feller property and uniform transience; and investigate the connection between the validity of these properties on the base space and its covering. For both graphs and manifolds, we prove the equivalence of stochastic incompleteness of the base and that of its cover. Along the way we also give some new conditions for the Feller property to hold on graphs., 28 pages

In this chapter we extend the concepts of Orientability, Differential Form with Compact Support, Integration and Poincare Adjunctions from manifolds to fiber bundles with the aim of extending the definition of Poincare Duality to make duality be relative to a base space.

In this study, we analyze statistical inference based on the Wasserstein geometry in the case that the base space is one-dimensional. By using the location-scale model, we derive the W-estimator that explicitly minimizes the transportation cost from the empirical distribution to a statistical model and study its asymptotic behaviors. We show that the W-estimator is consistent and explicitly give its asymptotic distribution by using the functional delta method. The W-estimator is Fisher efficient in the Gaussian case.

In this paper, a gait generation method for redundant under-actuated planar snake-like robots is developed. The proposed method generates gaits that guarantee achieving the desired net rotation of a floating snake-robot, that is, in the fiber space. This is done by specifying closed trajectories in the actuated base space, that is, cyclic time trajectories for all the actuated degrees of freedom. The proposed method is validated for purely mechanical systems that have a three-dimensional base space. Intuition is shed on how the proposed method could be modified to be applicable to higher-dimensional base spaces.

The existence of a strict deformation quantization of $X_k=S(M_k({\mathbb{C}}))$, the state space of the $k\times k$ matrices $M_k({\mathbb{C}})$ which is canonically a compact Poisson manifold (with stratified boundary) has recently been proven by both authors and K. Landsman \cite{LMV}. In fact, since increasing tensor powers of the $k\times k$ matrices $M_k({\mathbb{C}})$ are known to give rise to a continuous bundle of $C^*$-algebras over $I=\{0\}\cup 1/\mathbb{N}\subset[0,1]$ with fibers $A_{1/N}=M_k({\mathbb{C}})^{\otimes N}$ and $A_0=C(X_k)$, we were able to define a strict deformation quantization of $X_k$ \`{a} la Rieffel, specified by quantization maps $Q_{1/N}: \tilde{A}_0\rightarrow A_{1/N}$, with $\tilde{A}_0$ a dense Poisson subalgebra of $A_0$. A similar result is known for the symplectic manifold $S^2\subset\mathbb{R}^3$, for which in this case the fibers $A'_{1/N}=M_{N+1}(\mathbb{C})\cong B(\text{Sym}^N(\mathbb{C}^2))$ and $A_0'=C(S^2)$ form a continuous bundle of $C^*$-algebras over the same base space $I$, and where quantization is specified by (a priori different) quantization maps $Q_{1/N}': \tilde{A}_0' \rightarrow A_{1/N}'$. In this paper we focus on the particular case $X_2\cong B^3$ (i.e the unit three-ball in $\mathbb{R}^3$) and show that for any function $f\in \tilde{A}_0$ one has $\lim_{N\to\infty}||(Q_{1/N}(f))|_{\text{Sym}^N(\mathbb{C}^2)}-Q_{1/N}'(f|_{_{S^2}})||_N=0$, were $\text{Sym}^N(\mathbb{C}^2)$ denotes the symmetric subspace of $(\mathbb{C}^2)^{N \otimes}$. Finally, we give an application regarding the (quantum) Curie-Weiss model., Comment: 27 pages no figures, minor changes, accepted for publication in Letters in Mathematical Physics

A review of trends in space robotics identifies a promising class of assembly and service robotic space modules (ASRSM). This paper considers the topical problem of automating the assembly of large space objects. For certain ASRSM operations that are of practical interest, engineers can apply a design model comprising a moving base, a handler, and a payload. A massless single-DoF handler is an important special case. It is shown that in the absence of forces external to the system, the payload displacement with respect to the base can be described by an independent differential equation derived by Routhian formalism. In the common case where the angular momentum of the original system is non-zero, this equation is the dynamics equation for some non-linear oscillatory system, which, using the traditional terminology of analytical mechanics, we referred to as a reduced system and is seen as a controllable object. The proposed approach is convenient for studying the Eigen movement of a system by the internal degree of freedom, which movement is caused by the non-zero values both on the (neglected) cyclic axis, and on the hinged positional axis. Using these movements to synthesize the controls makes sense from the point of view of an important principle applicable to robotic movement arrangements; the principle consists in harmonizing the free and forced movements of the handler. The paper considers a modeling problem, which is of practical interest in its own right. The results can be generalized to apply to a general case of spatial motion of a system.

Recent results of arXiv:1907.08788 on universal black holes in $d$ dimensions are summarized. These are static metrics with an isotropy-irreducible homogeneous base space which can be consistently employed to construct solutions to virtually any metric theory of gravity in vacuum., 4 pages. Short summary of arXiv:1907.08788. Presented at the 6th Conference of the Polish Society on Relativity, Szczecin, Poland, September 23--26, 2019

In this work we study the following three space problem for operator spaces: if X is an operator space with base space isomorphic to a Hilbert space and X contains a completely isomorphic copy of the operator Hilbert space OH with respective quotient also completely isomorphic to OH, must X be completely isomorphic to OH? This problem leads us to the study of short exact sequences of operator spaces, more specifically those induced by complex interpolation, and their splitting. We show that the answer to the three space problem is negative, giving two different solutions.

We show that the homology of ordered configuration spaces of finite trees with loops is torsion free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete generating set for all homology groups of configuration spaces of trees with loops and the first homology group of configuration spaces of general finite graphs. An important technique in the paper is the identification of the $E^1$-page and differentials of Mayer-Vietoris spectral sequences for configuration spaces., Comment: 21 pages, 7 figures. Revised Proposition 2.3, other minor changes

It is know that the Valdivia compact spaces can be characterized by a special family of retractions called r-skeleton (see [10] ). Also we know that there are compact spaces with r-skeletons which are not Valdivia. In this paper, we shall study r-skeletons and special families of closed subsets of compact spaces. We prove that if X is a zero-dimensional compact space and { r s : s ∈ Γ } is an r-skeleton on X such that | r s ( X ) | ≤ ω for all s ∈ Γ , then X has a dense subset consisting of isolated points. Also we give conditions to an r-skeleton in order that this r-skeleton can be extended to an r-skeleton on the Alexandroff Duplicate of the base space. The standard definition of a Valdivia compact spaces is via a Σ-product of a power of the unit interval. Following this fact we introduce the notion of π-skeleton on a compact space X by embedding X in a suitable power of the unit interval together with a pair ( F , φ ) , where F is family of metric separable subspaces of X and φ an ω-monotone function which satisfy certain properties. This new notion generalize the idea of a Σ-product. We prove that a compact space admits a r-skeleton iff it admits a π-skeleton. This equivalence allows to give a new proof of the fact that the product of compact spaces with r-skeletons admits an r-skeleton (see [9] ).

This paper studies dispersiveness of semiflows on fiber bundles. The main result says that a right invariant semiflow on a fiber bundle is dispersive on the base space if and only if there is no almost periodic point and the semiflow is dispersive on the total space. A special result states that linear semiflows on vector bundles are not dispersive.

For each natural number n >= 4, we determine the unique lowest volume hyperbolic 3-orbifold whose torsion orders are bounded below by n. This lowest volume orbifold has base space the 3-sphere and singular locus the figure-8 knot, marked n. We apply this result to give sharp lower bounds on the volume of a hyperbolic manifold in terms of the order of elements in its symmetry group., 17 pages, 1 figure. v2 contains minor edits. To appear in Transactions of the AMS

We define a new invariant $w$ for hyperelliptic Lefschetz fibrations over closed oriented surfaces, which counts the number of Dirac braids included intrinsically in the monodromy, by using chart description introduced by the second author. As an application, we prove that two hyperelliptic Lefschetz fibrations of genus $g$ over a given base space are stably isomorphic if and only if they have the same numbers of singular fibers of each type and they have the same value of $w$ if $g$ is odd. We also give examples of pair of hyperelliptic Lefschetz fibrations with the same numbers of singular fibers of each type which are not stably isomorphic.

In this paper, we define slant helices in the three-dimensional anti-De Sitter space $$\mathbb {H}^{3}_{1}$$ and give characterizations in terms of the curvatures of the curve. We also introduce helix surfaces in $$\mathbb {H}^{3}_{1}$$ and, after proving that every helix surface is a flat surface, we give characterizations of helix surfaces based on the geometric model of pseudo-quaternions for $$\mathbb {H}^{3}_{1}$$ . These characterizations, with the support of Hopf fibrations, allow us to build helix surfaces in $$\mathbb {H}^{3}_{1}$$ from two curves in the base space ( $$\mathbb {H}^{2}$$ or $$\mathbb {S}^{2}_{1}$$ ), one of them being a general helix. Finally, we obtain a geometric integration of slant helices by proving that these curves can be viewed as geodesics (or other special curves) in certain flat surfaces of $$\mathbb {H}^{3}_{1}$$ . This gives us a method to geometrically construct all the slant helices in $$\mathbb {H}^{3}_{1}$$ .

We construct six-dimensional (6D) F-theory models in which discrete $\mathbb{Z}_5, \mathbb{Z}_4, \mathbb{Z}_3,$ and $\mathbb{Z}_2$ gauge symmetries arise. We demonstrate that a special family of "Fano 3-folds" is a useful tool for constructing the aforementioned models. The geometry of Fano 3-folds in the constructions of models can be useful for understanding discrete gauge symmetries in 6D F-theory compactifications. We argue that the constructions of the aforementioned models are applicable to Calabi-Yau genus-one fibrations over any base space, except models with a discrete $\mathbb{Z}_5$ gauge group. We construct 6D F-theory models with a discrete $\mathbb{Z}_5$ gauge group over the del Pezzo surfaces, as well as over $\mathbb{P}^1\times\mathbb{P}^1$ and $\mathbb{P}^2$. We also discuss some applications to four-dimensional F-theory models with discrete gauge symmetries., 19 pages, minor changes and minor typos corrected

Smooth tropical cubic surfaces are parametrized by maximal cones in the unimodular secondary fan of the triple tetrahedron. There are $344\, 843 \,867$ such cones, organized into a database of $14\,373\,645$ symmetry classes. The Schl\"afli fan gives a further refinement of these cones. It reveals all possible patterns of lines on tropical cubic surfaces, thus serving as a combinatorial base space for the universal Fano variety. This article develops the relevant theory and offers a blueprint for the analysis of big data in tropical geometry., Comment: Major revision: improved presentation and Section 7 deleted

We study cosymplectic-like statistical submersions. It is shown that for a cosymplectic-like statistical submersion, the base space is a Kahler-like statistical manifold and each fiber is a cosymplectic-like statistical manifold. We find the characterizations of the total and the base spaces under certain conditions. Examples of cosymplectic-like statistical manifolds and their submersions are also given.

We consider quantum symmetric algebras, FRT bialgebras and, more generally, intertwining algebras for pairs of Hecke symmetries which represent quantum hom-spaces. The paper makes an attempt to investigate Koszulness and Gorensteinness of those graded algebras without a restriction on the parameter q of the Hecke relation used earlier. When q is a root of 1, positive results require a restriction on the indecomposable modules for the Hecke algebras of type A that can occur as direct summands of representations in the tensor powers of the base space., Comment: plain TeX

We introduce conformal anti-invariant submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations which are arisen from the definition of a conformal submersion and find necessary and sufficient conditions for a conformal anti-invariant submersion to be totally geodesic. We also check the harmonicity of such submersions and show that the total space has certain product structures. Moreover, we obtain curvature relations between the base space and the total space, and find geometric implications of these relations., 41 pages, Submitted. arXiv admin note: text overlap with arXiv:1302.5108 by other authors

Pullback dynamics of nonautonomous dynamical systems has been considerably developed. However, it is still a tough job to study forward dynamics of nonautonomous dynamical systems, since forward attractors were only obtained in some particular cases. In the paper, under some reasonable conditions, it is shown that closing to a local pullback attractor, there is an approximate forward attractor. Specifically, let ϕ be a cocycle semiflow on a Banach space X with driving system θ on a base space P. Suppose that the base space P is compact and ϕ is uniformly asymptotically compact. Let A(∙) be a local pullback attractor with being compact. We prove that every e-extended neighborhood Ae(∙) of A(∙) will forward attract every bounded set B(∙) that is pullback attracted by A(∙). We then call Ae(∙) an approximate forward attractor of ϕ.

This paper extends the results from the author's previous paper to consider finite, fiber- and orientation- preserving group actions on closed, orientable Seifert manifolds $M$ that fiber over a non-orientable base space. An orientable base space double cover $\tilde{M}$ of $M$ is constructed and then an isomorphism between the fiber- and orientation-preserving diffeomorphisms of ${M}$ and the fiber- and orientation-preserving actions on $\tilde{M}$ that preserve the orientation on the fibers and commute with the covering translation is shown. This result and previous results lead to a construction of all actions that satisfy a condition on the obstruction term and the structure of the finite groups that can act on $M$., 9 pages, 4 figures. Final revision before publication

Publisher Summary This chapter focuses on eta invariants and automorphisms of compact complex manifolds. It also presents the recent existence results of Einstein–Kahler metrics of a positive Ricci curvature. By an automorphism of a vector bundle is meant a diffeomorphism of the total space such that it descends to a diffeomorphism of the base space and that it maps a fiber to a fiber isomorphically.

We establish the factorization of Dirac operators on Riemannian submersions of compact spin$^c$ manifolds in unbounded KK-theory. More precisely, we show that the Dirac operator on the total space of such a submersion is unitarily equivalent to the tensor sum of a family of Dirac operators with the Dirac operator on the base space, up to an explicit bounded curvature term. Thus, the latter is an obstruction to having a factorization in unbounded KK-theory. We show that our tensor sum represents the bounded KK-product of the corresponding KK-cycles and connect to the early work of Connes and Skandalis., Comment: 26 pages

Let$Q\rightarrow X$be a continuous principal bundle whose group$G$is reductive. A flow${\it\phi}$of automorphisms of$Q$endowed with an ergodic probability measure on the compact base space$X$induces two decompositions of the flag bundles associated to$Q$: a continuous one given by the finest Morse decomposition and a measurable one furnished by the multiplicative ergodic theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under perturbations leaving unchanged the flow on the base space.

This paper provides explicit closed formulas in terms of tautological classes for the cycle classes of the height and Artin invariant strata in families of K3 surfaces. The proof is uniform for all strata and uses a flag space as the computations in [arXiv:math/0412272] for the Ekedahl-Oort strata for families of abelian varieties, but employs a Pieri formula formula to determine the push down to the base space., 34 pages, latex; version revised by second author; to appear in Selecta Math

Almost contact and almost complex structures in thetangent bundle have been studied by K. Yano and S. Ishihara[5] andothers. In the present paper, we have studied Lorentzian almost r -para-contact structure in the tangent bundle. Some results relatedto Lie-derivative have been studied. 1. IntroductionLet M be a n -dimensional difierentiable manifold of C 1 class and TS p ( M ) the tangent space of M at a point p of M . Then the set T ( M ) = p2M T p ( M ) is called the tangent bundle over the manifold M . For anypoint ~ p of T ( M ), the correspondence ~ p ! p determines the bundle pro-jection … : T ( M ) ! M . Thus … (~ p ) = p , where … : T ( M ) ! M deflnesthe bundle projection of T ( M ) over M . The set … i 1 ( p ) is called the flbre over p 2 M and M the base space .Vertical lifts:If f is a function in M , then we write f V for the function in T ( M )obtained by forming the composition of … : T ( M ) ! M and f : M ! R so that f V = f –… . Thus, if a point ~ p 2 … i 1 ( U ) has induced coordinates(

As policy for regional development in bottom-up style is introduced, each local government reflects opinion of regional residents and experts, and continues to strive for active use of regional capability and natural resources. As a result, there are active movements for regional development in Eup and Myeon unit or village unit inside local government. Comprehensive arrangement project in areas of Eup and Myeon is proceeded with a goal of improvement of life quality for regional residents through strengthening base function of Eup and Myeon areas and improving function of basic service by expanding facility of optimal level available to an unspecified number of the general public such as educational, cultural, welfare facility etc. in Eup and Myeon areas which are base space of rural communities. For analysis method of region for selecting areas where comprehensive arrangement project is done, this study suggested analysis of connection structure based on interaction and analysis of centrality. And empirical analysis was carried out with Buyeo province in Chungcheongnamdo.

Let Aut ( p ) denote the space of all self-fibre-homotopy equivalences of a principal U n ( C ) -bundle ξ : U n ( C ) → E → p X of simply connected CW complexes with E finite. We show that there exists an inequality: n − N ( p ) ⩽ Hnil Q ( Aut ( p ) 0 ) ⩽ n for any base space X, where N ( p ) = min { k | k = card ( i | χ 2 i ≠ 0 , i n ) } with χ 2 i denoting the generalized characteristic class of the bundle p, and Hnil Q ( Aut ( p ) 0 ) = n − 1 when X = C P m with m ⩾ n − 2 and p is not rational trivial with mild restriction. We also show that Hnil Q ( Aut ( p ) 0 ) = n if p is a rational fibre-homotopy trivial bundle and X is finite.

This is a largely expository account of various aspects of the Borsuk–Ulam theorem, including extensions of the classical theorem to families of maps parametrized by a base space and to multivalued maps. The main technical tool is the Euler class with compact supports.

We formulate several criteria under which the symmetries associated with the Killing and Killing-Yano tensors on the base space can be lifted to the symmetries of the full warped geometry. The procedure is explicitly illustrated on several examples, providing new prototypes of spacetimes admitting such tensors. In particular, we study a warped product of two Kerr-NUT-(A)dS spacetimes and show that it gives rise to a new class of highly symmetric vacuum (with cosmological constant) black hole solutions that inherit many of the properties of the Kerr-NUT-(A)dS geometry., 10 pages, no figures

In this chapter we prove maximal Lp-regularity for various linear parabolic and elliptic problems. These results will be crucial for our study of quasilinear parabolic problems, including those introduced in Chapter 1.

The aim of this paper is to describe the obstruction for an almost Lagrangian fibration to be Lagrangian, a problem which is central to the classification of Lagrangian fibrations and, more generally, to understanding the obstructions to carry out surgery of integrable systems, an idea introduced by Zung. It is shown that this obstruction (namely, the homomorphism of Dazord and Delzant) is related to the cup product in cohomology with local coefficients on the base space B of the fibration. The map is described explicitly and some examples are calculated, thus providing the first examples of non-trivial Lagrangian obstructions., 17 pages, to appear in Diff. Geom. Appl

In “Rips complexes and covers in the uniform category” (Brodskiy et al., preprint [4] ) the authors define, following James (1990) [5] , covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given. This paper extends these results by investigating the existence of these covering maps relative to subgroups of the uniform fundamental group and the fundamental group of the base space.

We consider a degeneration of genus 2 curves, which is op- posite to maximal degeneration in a sense. Such a degeneration of curves yields a variation of mixed Hodge structure with monodromy weight fil- tration. The mixed Hodge structure at each fibre, which is dierent from the limit mixed Hodge structure of Schmid and Steenbrink, can be real- ized as H1 of a noncompact singular elliptic curve. We also prove that the pull back of the above variation of mixed Hodge structure to a dou- ble cover of the base space comes from a family of noncompact singular elliptic curves.

This paper shows that for mechanical systems, the dimension of whose base space is larger than time (there also exist spatial coordinates), the system of equations defining the evolution of the system must be a hyperbolic system of pseudodifferential equations.

Let RP(k) denotethe k-dimensional real projective space. This article deter mines which cobordism classes are represented by the total space of a fibering with pre scribed base space RP (3) × RP (1), RP (2) × RP (1), RP (2) × RP (1) × RP (1) or RP (3) × RP (2).

We show that the universal covering [math] of any compact, negatively curved manifold [math] has an exponential growth rate which is strictly greater than the exponential growth rate of any other normal covering [math] . Moreover, we give an explicit formula estimating the difference between [math] and [math] in terms of the systole of [math] and of other elementary geometric parameters of the base space [math] . Then we discuss some applications of this formula to periodic geodesics, to the bottom of the spectrum and to the critical exponent of normal coverings.

In the present paper we consider Riemannian coverings $(X,g) \to (M,g)$ with residually finite covering group $\Gamma$ and compact base space $(M,g)$. In particular, we give two general procedures resulting in a family of deformed coverings $(X,g_\eps) \to (M,g_\eps)$ such that the spectrum of the Laplacian $\Delta_{(X_\eps,g_\eps)}$ has at least a prescribed finite number of spectral gaps provided $\eps$ is small enough. If $\Gamma$ has a positive Kadison constant, then we can apply results by Br\"uning and Sunada to deduce that $\spec \Delta_{(X,g_\eps)}$ has, in addition, band-structure and there is an asymptotic estimate for the number $N(\lambda)$ of components of $\spec {\laplacian {(X,g_\eps)}}$ that intersect the interval $[0,\lambda]$. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results., Comment: final version (26 pages, 2 figures). to appear in Rev. Math. Phys

We prove the Herbert multiple-point formula modulo 2-torsion using the method of the universal immersion.

Let $f,g:\cc^2\ra\cc$ be two quasi-homogeneous polynomials. We compute the $V$-filtration of the restriction of $f$ to any plane curve $C_t=g^{-1}(t)$ and show that the Gorenstein generator $dx\wedge dy/dg$ is a primitive form. Using results of A. Douai and C. Sabbah, we conclude that base space of the miniversal unfolding of $f_t:=f|_{C_t}$ is a Frobenius manifold. At the singular fibre $C_0$ we obtain a non-massive Frobenius manifold., 11 pages

Given a Waelbroeck ring R, we prove that the Grassmannian of a projective finitely generated R -module is a topological manifold modeled on a topological abelian group of R -linear maps. Fibre bundles of fibre type a module as above, over a compact base space B, admitting R -valued partitions of unity, are classified by the homotopy classes of continuous maps on B with values in the respective Grassmannian. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We show that a diffeological bundle gives rise to an exact sequence of internal tangent spaces. We then introduce two new classes of diffeological spaces, which we call weakly filtered and filtered diffeological spaces, whose tangent spaces are easier to understand. These are the diffeological spaces whose categories of pointed plots are (weakly) filtered. We extend the exact sequence one step further in the case of a diffeological bundle with filtered total space and base space. We also show that the tangent bundle $T^H X$ defined by Hector is a diffeological vector space over $X$ when $X$ is filtered or when $X$ is a homogeneous space, and therefore agrees with the dvs tangent bundle introduced by the authors in a previous paper., v3: new results and improvements to exposition; 14 pages; this version to appear in Proceedings of the AMS