Quotient Map Research Articles

Stabilizability preserving quotients and partial feedback linearization

This paper presents the problem of stabilizability preserving quotients. Given a control system and its quotient the problem of stabilizability preserving quotients seeks to characterize all the quotients for which the system is stabilizable if and only if the quotient is stabilizable. This paper presents a mathematical formulation of this problem using the language of differential geometry. Taking partial feedback linearization as an instance of quotienting of a control system, it is shown that quotients constructed via partial feedback linearization are stabilizability preserving if the “zero dynamics” are stable. In addition to this sufficient condition another result developed in this paper is the development of a construction of “zero dynamics” that makes us of Ehresmann connections. Making use of the fiber bundle structure induced by the quotient map it is shown that it is possible to define an Ehresmann connection on the state manifold such that the “zero dynamics” can be defined as a vertical vector field that is related to the horizontal lift of the linear system.

On the quotients of c-spaces

In this paper, the concept of weak c-structure generated by a family of functions is introduced and quotient spaces are introduced as a particular case of this. Properties of quotient maps are explored. A method of finding quotient space of topologizable and graphical c-spaces are described.

Ordered groupoid quotients and congruences on inverse semigroups

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s $${\mathcal {J}}$$ –relation. The corresponding equivalence relation $$\simeq _N$$ is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.

Khovanov–Seidel quiver algebras and Ozsváth–Szabó's bordered theory

In the 2016 preprint “Kauffman states, bordered algebras, and a bigraded knot invariant,” Ozsváth and Szabó introduced a set of algebraic constructions in the spirit of bordered Heegaard Floer homology. Their constructions can be used to compute knot Floer homology algebraically for knots in the 3-sphere. In this paper we investigate a relationship between Ozsváth–Szabó's bordered theory and the algebras and bimodules constructed by Khovanov and Seidel in “Quivers, Floer cohomology, and braid group actions” (2002). Specifically, we show that the Khovanov–Seidel quiver algebras are isomorphic to quotients of idempotent truncations of some of Ozsváth–Szabó's algebras. Furthermore, we show that the dg bimodule associated to a braid generator by Khovanov–Seidel, with the right action restricted using the quotient map, is homotopy equivalent to Ozsváth–Szabó's DA bimodule with the left action induced using the quotient map.

Cross sections and pseudo-homomorphisms of topological abelian groups

We say that a mapping ω between two topological abelian groups G and H is a pseudo-homomorphism if the associated map (x,y)∈G×G↦ω(x+y)−ω(x)−ω(y)∈H is continuous. This notion appears naturally in connection with cross sections (continuous right inverses for quotient mappings): given an algebraically splitting, closed subgroup H of a topological group X such that the projection π:X→X/H admits a cross section, one obtains a pseudo-homomorphism of X/H to H, and conversely. We show that H splits as a topological subgroup if and only if the corresponding pseudo-homomorphism can be decomposed as a sum of a homomorphism and a continuous mapping. We also prove that pseudo-homomorphisms between Polish groups satisfy the closed graph theorem. Several examples are given.

Homotopy principles for equivariant isomorphisms

Let G G be a reductive complex Lie group acting holomorphically on Stein manifolds X X and Y Y . Let p X : X → Q X p_X\colon X\to Q_X and p Y : Y → Q Y p_Y\colon Y\to Q_Y be the quotient mappings. When is there an equivariant biholomorphism of X X and Y Y ? A necessary condition is that the categorical quotients Q X Q_X and Q Y Q_Y are biholomorphic and that the biholomorphism φ \varphi sends the Luna strata of Q X Q_X isomorphically onto the corresponding Luna strata of Q Y Q_Y . Fix φ \varphi . We demonstrate two homotopy principles in this situation. The first result says that if there is a G G -diffeomorphism Φ : X → Y \Phi \colon X\to Y , inducing φ \varphi , which is G G -biholomorphic on the reduced fibres of the quotient mappings, then Φ \Phi is homotopic, through G G -diffeomorphisms satisfying the same conditions, to a G G -equivariant biholomorphism from X X to Y Y . The second result roughly says that if we have a G G -homeomorphism Φ : X → Y \Phi \colon X\to Y which induces a continuous family of G G -equivariant biholomorphisms of the fibres p X − 1 ( q ) p_X{^{-1}}(q) and p Y − 1 ( φ ( q ) ) p_Y{^{-1}}(\varphi (q)) for q ∈ Q X q\in Q_X and if X X satisfies an auxiliary property (which holds for most X X ), then Φ \Phi is homotopic, through G G -homeomorphisms satisfying the same conditions, to a G G -equivariant biholomorphism from X X to Y Y . Our results improve upon those of our earlier paper [J. Reine Angew. Math. 706 (2015), 193–214] and use new ideas and techniques.

On statistically sequentially quotient maps

In this paper, we introduce the concept of statistically sequentially quotient map which is a generalization of sequence covering map and discuss the relation with covering maps by some examples. Using this concept, we give an affirmative answer for a question by Fucai Lin and Shou Lin.

Topology of unitary groups and the prime orders of binomial coefficients

Let $c:SU(n)\rightarrow PSU(n)=SU(n)/\mathbb{Z}_{n}$ be the quotient map of the special unitary group $SU(n)$ by its center subgroup $\mathbb{Z}_{n}$. We determine the induced homomorphism $c^{\ast}:$ $H^{\ast}(PSU(n))\rightarrow H^{\ast}(SU(n))$ on cohomologies by computing with the prime orders of binomial coefficients

Quotient manifolds of flows

This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of countably presented groups is induced by the quotient map of some flow, and that higher homology can also be controlled. Manifolds of fixed dimension arising as quotients of flows on Euclidean space realize all even (and some odd) intersection pairings, and all homotopy spheres of dimension at least two arise in this manner. Most Euclidean spaces of dimensions five and higher have families of topologically equivalent but smoothly inequivalent flows with quotient homeomorphic to a manifold with flexibly chosen homology. For [Formula: see text], there is a topological flow on (ℝ2r+1 − 8 points) × ℝm that is unsmoothable, although smoothable near each orbit, with quotient an unsmoothable topological manifold.

Dynamics of skew products of interval maps

In the study of dynamical systems in the class of skew products, the present paper involves the direction most closely connected with advances in one-dimensional dynamics. The main results obtained over the last decades on the dynamics of skew products of interval maps are surveyed. Included here are new results on the structure of the non-wandering set and the centre for -smooth skew products of interval maps that are endomorphisms whose quotient maps have complicated dynamics. These results are used to describe the space of skew products of this type. Bibliography: 125 titles.

Convexity for twisted conjugation

Let $G$ be a compact, simply connected Lie group. If $\mathcal{C}_1,\mathcal{C}_2$ are two $G$-conjugacy classes, then the set of elements in $G$ that can be written as products $g=g_1g_2$ of elements $g_i\in \mathcal{C}_i$ is invariant under conjugation, and its image under the quotient map $G\to G/\operatorname{Ad}(G)$ is a convex polytope inside the Weyl alcove. In this note, we will prove an analogous statement for twisted conjugations relative to group automorphisms. The result will be obtained as a special case of a convexity theorem for group-valued moment maps which are equivariant with respect to the twisted conjugation action.

Large social inequalities and low levels of socio-economic segregation in Vilnius

The city of Vilnius has experienced major shifts in occupational structure between 2001 and 2011 and at the same time there were major transitions in the housing market and suburbanization. The main aim of this chapter is to get more insight in recent socio-economic segregation processes in Vilnius. We used occupational groups as a proxy for socio-economic status, and census tract level data to measure segregation in Vilnius and its three main housing zones during 2001 and 2011. Notwithstanding the major economic and social changes of post-communist society, we found low levels of segregation and modest change during the last decade. Local patterns of segregation were explored using location quotient maps. The analyses illustrated a deepening social divide in the city between the relatively rich north and the poorer south of the city, but the inner city changes are somehow ambiguous. In this chapter we argue that the main factors of socio-spatial change in Vilnius are related to an exceptionally high share of housing estates in the city and the polycentric urban system of the country. Together with 'fast-track' reforms after 1990 this urban system gave a unique character to the current processes and patterns of segregation.

Singular twisted sums generated by complex interpolation

We present new methods to obtain singular twisted sums X ⊕ Ω X X\oplus _\Omega X (i.e., exact sequences 0 → X → X ⊕ Ω X → X → 0 0\to X\to X\oplus _\Omega X \to X\to 0 in which the quotient map is strictly singular) when X X is an interpolation space arising from a complex interpolation scheme and Ω \Omega is the induced centralizer. Although our methods are quite general, we are mainly concerned with the choice of X X as either a Hilbert space or Ferenczi’s uniformly convex Hereditarily Indecomposable space. In the first case, we construct new singular twisted Hilbert spaces (which includes the only known example so far: the Kalton-Peck space Z 2 Z_2 ). In the second case we obtain the first example of an H.I. twisted sum of an H.I. space. During our study of singularity we introduce the notion of a disjointly singular twisted sum of Köthe function spaces and construct several examples involving reflexive p p -convex Köthe function spaces (which includes the function space version of the Kalton-Peck space Z 2 Z_2 ). We then use Rochberg’s description of iterated twisted sums to show that there is a sequence F n \mathcal F_n of H.I. spaces so that F m + n \mathcal F_{m+n} is a singular twisted sum of F m \mathcal F_m and F n \mathcal F_n , while for l > n l>n the direct sum F n ⊕ F l + m \mathcal F_n \oplus \mathcal F_{l+m} is a nontrivial twisted sum of F l \mathcal F_l and F m + n \mathcal F_{m+n} .

Adjoint quotient maps for the restricted Lie algebras Wn and Sn

ABSTRACTWe give an explicit description of the adjoint quotient maps for the Jacobson–Witt algebra Wn and the special algebra Sn. An analog of Kostant’s differential criterion of regularity is given for Wn. Furthermore, we describe the fiber of the adjoint quotient map for Sn and construct analogs of Kostant’s transverse slice for Wn. In addition, we analyze nilpotent elements and generalize some of Premet’s results ([8]) to Sn.

Nonwandering Sets of C 1-Smooth Skew Products of Interval Maps with Complicated Dynamics of Quotient Map

We obtain exact formulas describing the nonwandering set of a C 1-smooth skew product of interval maps with Ω-stable quotient map of type ≻ 2∞.

The quotient map on the equivariant Grothendieck ring of varieties

For a scheme S with a good action of a finite abelian group G having enough roots of unity we show that the quotient map on the G-equivariant Grothendieck ring of varieties over S is well defined with image in the Grothendieck ring of varieties over S/G in the tame case, and in the modified Grothendieck ring in the wild case. To prove this we use a result on the class of the quotient of a vector space by a quasi-linear action in the Grothendieck ring of varieties due to Esnault and Viehweg, which we also generalize to the case of wild actions. As an application we deduce that the quotient of the motivic nearby fiber is a well defined invariant.

Basic Forms and Orbit Spaces: a Diffeological Approach

If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the identity component acts properly, where on the quotient space we take differential forms in the diffeological sense.

Property T of group homomorphisms

In this article, we define and study property T for continuous homomorphisms between topological groups. If G is a locally compact group, we show that λG:G→U(L2(G)) has property T if and only if either G is compact or G is non-amenable. Moreover, the abelianization Gab:=G/[G,G]‾ is compact if and only if every continuous homomorphism from G to any abelian topological group has property T. Moreover, we show that G has property (T,FD) if and only if any continuous homomorphism from G to any compact group has property T. In the case when G is almost connected, the above is also equivalent to the canonical map from G to its Bohr compactification being a quotient map. We also give some new equivalent forms of the strong property T of a locally compact group. As a consequence, if G is a second countable and has strong property T and H is a closed subgroup of G, there exist at most one G-invariant mean on G/H.

Realization of compact spaces as cb-Helson sets

We show that, given a compact Hausdorff space $\Omega$, there is a compact group ${\mathbb G}$ and a homeomorphic embedding of $\Omega$ into ${\mathbb G}$, such that the restriction map ${\rm A}({\mathbb G})\to C(\Omega)$ is a complete quotient map of operator spaces. In particular, this shows that there exist compact groups which contain infinite cb-Helson subsets, answering a question raised in [Choi--Samei, Proc. AMS 2013; cf. http://arxiv.org/abs/1104.2953]. A negative result from the same paper is also improved.

Multivalued functions and nonwandering set of skew products of maps of an interval with complicated dynamics of quotient map

With the use of specialmultivalued functions, we give a description of nonwandering set of C 1-smooth skewproducts of maps of an interval with Ω-stable quotient map of the type ≻ 2∞.