Kurdyka–Łojasiewicz Functions and Mapping Cylinder Neighborhoods

Let X be a closed subset of a manifold without boundary Q. A codimension 0 submanifold M of Q is a mapping cylinder neighborhood of X if it is a closed neighborhood of X in Q and if there is a proper map r: AM-P X such that M is homeomorphic to the mapping cylinder of r, fixing AM and X in the natural way. For example, regular neighborhoods of locally finite complexes in PL manifolds are mapping cylinder neighborhoods. We do not insist that M be collared in Q since this can always be arranged by an isotopy of M into itself along the structure lines. Edwards ([5]) contains a lot of information about mapping cylinder neighborhoods. It follows immediately from the definition that subsets possessing mapping cylinder neighborhoods are retracts of the neighborhoods, hence are finite dimensional ANR's. The converse question, whether finite dimensional ANR's have mapping cylinder neighborhoods, at least for some embeddings in manifolds, is interesting. If they do, then compact, finite dimensional ANR's have finite homotopy type, since these neighborhoods are compact manifolds, and so have finite homotopy type by Kirby-Siebenmann (17]). We prove a partial result in the direction of the converse, namely, if X is a locally compact ANR, embedded as a closed, 1-LC codimension 4 subset of a finite dimensional manifold without boundary Q, then X x M, has a mapping cylinder neighborhood in Q x M, where either M. M= SI, or (M, Mj) = (R', [0, ao)). Our proof also works in infinite dimensions and yields the fact that if X is a locally compact ANR which is embedded as a closed Z-set in a Hilbert cube manifold Q, then X x M. has a cylinder neighborhood in Q x M. Taking suitable compactifications in the case (M, Mj) = (Ri, [0, ao)), one finds that the cone on X is the image of the cone on Q by a map with contractible point inverses. J. West ([10]) has taken this last form of our result and shown that in the infinite dimensional setting it implies X actually has a mapping cylinder

Let X be a closed subset of a manifold without boundary Q. A codimension 0 submanifold M of Q is a mapping cylinder neighborhood of X if it is a closed neighborhood of X in Q and if there is a proper map r: AM-P X such that M is homeomorphic to the mapping cylinder of r, fixing AM and X in the natural way. For example, regular neighborhoods of locally finite complexes in PL manifolds are mapping cylinder neighborhoods. We do not insist that M be collared in Q since this can always be arranged by an isotopy of M into itself along the structure lines. Edwards ([5]) contains a lot of information about mapping cylinder neighborhoods. It follows immediately from the definition that subsets possessing mapping cylinder neighborhoods are retracts of the neighborhoods, hence are finite dimensional ANR's. The converse question, whether finite dimensional ANR's have mapping cylinder neighborhoods, at least for some embeddings in manifolds, is interesting. If they do, then compact, finite dimensional ANR's have finite homotopy type, since these neighborhoods are compact manifolds, and so have finite homotopy type by Kirby-Siebenmann (17]). We prove a partial result in the direction of the converse, namely, if X is a locally compact ANR, embedded as a closed, 1-LC codimension 4 subset of a finite dimensional manifold without boundary Q, then X x M, has a mapping cylinder neighborhood in Q x M, where either M. M= SI, or (M, Mj) = (R', [0, ao)). Our proof also works in infinite dimensions and yields the fact that if X is a locally compact ANR which is embedded as a closed Z-set in a Hilbert cube manifold Q, then X x M. has a cylinder neighborhood in Q x M. Taking suitable compactifications in the case (M, Mj) = (Ri, [0, ao)), one finds that the cone on X is the image of the cone on Q by a map with contractible point inverses. J. West ([10]) has taken this last form of our result and shown that in the infinite dimensional setting it implies X actually has a mapping cylinder

Embeddings with mapping cylinder neighborhoods

We characterize those compact subsets of the plane which have mapping cylinder neighborhoods, describe the neighborhood closures, and show that such neighborhood closures are topologically unique. The proofs employ the notion of prime ends. We also show that if U U is a mapping cylinder neighborhood of a pointlike continuum in S 3 {S^3} , then U ¯ \overline U is a 3 3 -cell.

We deal with the minimum number of critical points of circular functions with respect to two different classes of functions. The first one is the whole class of smooth circular functions and, in this case, the minimum number is the so called circular \(\varphi\) -category of the involved manifold. The second class consists of all smooth circular Morse functions, and the minimum number is the so called circular Morse–Smale characteristic of the manifold. The investigations we perform here for the two circular concepts are being studied in relation with their real counterparts. In this respect, we first evaluate the circular \(\varphi\)-category of several particular manifolds. In Sect. 5, of more survey flavor, we deal with the computation of the circular Morse–Smale characteristic of closed surfaces. Section 6 provides an upper bound for the Morse–Smale characteristic in terms of a new characteristic derived from the family of circular Morse functions having both a critical point of index 0 and a critical point of index n. The minimum number of critical points for real or circle valued Morse functions on a closed orientable surface is the minimum characteristic number of suitable embeddings of the surface in \(\mathbb{R}^{3}\) with respect to some involutive distributions. In the last section we obtain a lower and an upper bound for the minimum characteristic number of the embedded closed surfaces in the first Heisenberg group with respect to its noninvolutive horizontal distribution.

Classical Morse theory is concerned with the critical points of a class of smooth proper functions f from a manifold Z to the real numbers, called Morse functions. For our generalization, we will let Z be a closed Whitney stratified space in some ambient smooth manifold M. We will need analogues for the notions of smooth function, critical point, and Morse function for this setting. A smooth function on Z will be a function which extends to a smooth function on M. A critical point of a smooth function f will be a critical point of the restriction of f to any stratum S of Z. A proper function f is called Morse if (1) its restriction to each stratum has only nondegenerate critical points, (2) its critical values are distinct, and (3) the differential of f at a critical point p in S does not annihilate any limit of tangent spaces to a stratum other than S. This third condition is a sort of nondegeneracy condition normal to the stratum. If Z is subanalytic (which includes the real and complex analytic cases), then the set of Morse functions forms an open dense subset of the space of smooth functions, and Morse functions are structurally stable, just as in the classical case [P1].

Regular neighborhoods have proved to be a very useful tool in the theory of PL manifolds. In this paper we want to make a very easy construction of regular neighborhoods in the topological category. F. E. A. Johnson [6] has constructs regular neighborhoods in the topological category, but only in the case of nonintersection with the boundary. R. D. Edwards [2] has announced a very general construction of regular neighborhoods; see also [3]. The present construction has the advantage of allowing a “relative” version , (Theorem 13), in the sense that if L is a complex, K is a subcomplex, and L is locally tamely embedded in a topological manifold V , then one may find a regular neighborhood ofK in V , intersecting L in a regular neighborhood of K in L, in the usual PL sense. This is used in [10] to prove embedding theorems for topological manifolds. In [11] we have a proof that the opposite procedure is possible; namely a spine of a topological manifold. We should emphasize that the regular neighborhoods we obtain are mapping cylinder neighborhoods; i. e. if K ⊂ N where N is a regular neighborhood of K, then there is a map π : ∂N → K such that N is homeomorphic to the mapping cylinder of π (Theorem 15). Let K be a compact topological space with a given simple homotopy structure; i. e. of the homotopy type of a finite CW-complex, with the homotopy equivalence specified up to torsion.

Abstraet. The majority of results in computational learning theory are concerned with concept learning, i.e. with the special case of function learning for classes of functions with range {0, 1}. Much less is known about the theory of learning functions with a larger fange such as Nor IR. In particular relatively few results exist about he general structure of common models for function learning, and there are only very few nontrivial function classes for which positive learning results have been exhibited in any of these models. We introduce in this paper the notion of a binaly branching adversary tree for function learning, which allows us to give a somewhat surprising equivalent characterization f the optimal learning cost for learning a class of real-valued functions (in terms of a max-min definition which does not invoive any "learning " model). Another general structural result of this paper elates the cost for learning a union of function classes to the learning costs for the individual function classes. Furthermore, we exhibit an efficient leaming algorithm for learning convex piecewise linear functions from Rd into IR. Previously, the class of linear functions from 1R d into R was the only class of functions with multi-dimensional domain that was known to be learnable within the rigorous framework of a formal model for on-line leaming. Finally we give a sufficient condition for an arbitrary class 5 ~ of functions from IR into R that allows us to learn the class of all functions that can be written as the pointwise maximum of k functions from 5 r. This allows us to exhibit a number of further nontrivial classes of functions from ~ into R for which there exist eflicient]earning algorithms.

The majority of results in computational learning theory are concerned with concept learning, i.e. with the special case of function learning for classes of functions with range l0, 1r. Much less is known about the theory of learning functions with a larger range such as {\Bbb N} or {\Bbb R}. In particular relatively few results exist about the general structure of common models for function learning, and there are only very few nontrivial function classes for which positive learning results have been exhibited in any of these models. We introduce in this paper the notion of a binary branching adversary tree for function learning, which allows us to give a somewhat surprising equivalent characterization of the optimal learning cost for learning a class of real-valued functions (in terms of a max-min definition which does not involve any “learning” model). Another general structural result of this paper relates the cost for learning a union of function classes to the learning costs for the individual function classes. Furthermore, we exhibit an efficient learning algorithm for learning convex piecewise linear functions from {\Bbb R}^d into {\Bbb R}. Previously, the class of linear functions from {\Bbb R}^d into {\Bbb R} was the only class of functions with multidimensional domain that was known to be learnable within the rigorous framework of a formal model for online learning. Finally we give a sufficient condition for an arbitrary class {\cal F} of functions from {\Bbb R} into {\Bbb R} that allows us to learn the class of all functions that can be written as the pointwise maximum of k functions from {\cal F}. This allows us to exhibit a number of further nontrivial classes of functions from {\Bbb R} into {\Bbb R} for which there exist efficient learning algorithms.

The majority of results in computational learning theory are concerned with concept learning, i.e. with the special case of function learning for classes of functions with range {0, 1}. Much less is known about the theory of learning functions with a larger range such as $$\mathbb{N}$$ or $$\mathbb{R}$$ . In particular relatively few results exist about the general structure of common models for function learning, and there are only very few nontrivial function classes for which positive learning results have been exhibited in any of these models. We introduce in this paper the notion of a binary branching adversary tree for function learning, which allows us to give a somewhat surprising equivalent characterization of the optimal learning cost for learning a class of real-valued functions (in terms of a max-min definition which does not involve any “learning” model). Another general structural result of this paper relates the cost for learning a union of function classes to the learning costs for the individual function classes. Furthermore, we exhibit an efficient learning algorithm for learning convex piecewise linear functions from $$\mathbb{R}^d $$ into $$\mathbb{R}$$ . Previously, the class of linear functions from $$\mathbb{R}^d $$ into $$\mathbb{R}$$ was the only class of functions with multidimensional domain that was known to be learnable within the rigorous framework of a formal model for online learning. Finally we give a sufficient condition for an arbitrary class $$\mathcal{F}$$ of functions from $$\mathbb{R}$$ into $$\mathbb{R}$$ that allows us to learn the class of all functions that can be written as the pointwise maximum of k functions from $$\mathcal{F}$$ . This allows us to exhibit a number of further nontrivial classes of functions from $$\mathbb{R}$$ into $$\mathbb{R}$$ for which there exist efficient learning algorithms.

Closed choice and a Uniform Low Basis Theorem

In this paper we study the singular vanishing-viscosity limit of a gradient flow in a finite dimensional Euclidean space, focusing on the so-called delayed loss of stability of stationary solutions. We find a class of time-dependent energy functionals and initial conditions for which we can explicitly calculate the first discontinuity time [Formula: see text] of the limit. For our class of functionals, [Formula: see text] coincides with the blow-up time of the solutions of the linearized system around the equilibrium, and is in particular strictly greater than the time [Formula: see text] where strict local minimality with respect to the driving energy gets lost. Moreover, we show that, in a right neighborhood of [Formula: see text], rescaled solutions of the singularly perturbed problem converge to heteroclinic solutions of the gradient flow. Our results complement the previous ones by Zanini [ Discrete Contin. Dyn. Syst. Ser. A 18 ( 2007 ), 657–675], where the situation we consider was excluded by assuming the so-called transversality conditions, and the limit evolution consisted of strict local minimizers of the energy up to a negligible set of times.

Borel, Lebesgue, and Hausdorff described all uniformly closed families of real-valued functions on a set T whose algebraic properties are just like those of the set of all continuous functions with respect to some open topology on T. These families turn out to be exactly the families of all functions measurable with respect to some σ-additive and multiplicative ensembles on T. The problem of describing all uniformly closed families of bounded functions whose algebraic properties are just like those of the set of all continuous bounded functions remained unsolved. In the paper, a solution of this problem is given with the help of a new class of functions that are uniform with respect to some multiplicative families of finite coverings on T. It is proved that the class of uniform functions differs from the class of measurable functions.

For $n \geqslant 7$ we describe an $(n - 1)$-sphere $\Sigma$ wildly embedded in the $n$-sphere yet every point of $\Sigma$ has arbitrarily small neighborhoods bounded by flat $(n - 1)$-spheres, each intersecting $\Sigma$ in an $(n - 2)$-sphere. Not only do these examples for large $n$ run counter to what can occur when $n = 3$, they also illustrate the sharpness of high-dimensional taming theorems developed by Cannon and Harrold and Seebeck. Furthermore, despite their wildness, they have mapping cylinder neighborhoods, which both run counter to what is possible when $n = 3$ and also partially illustrate the sharpness of another high-dimensional taming theorem due to Bryant and Lacher.

Mapping Cylinder Neighborhoods