Heuristic approach for minimizing the projection error in the integrated mapping

In this paper, we have developed and examined a heuristic approach for minimizing the projection error in Sammon’s mapping applied in combination with the self-organizing map (SOM). As a final result, we need to visualize the neurons-winners of the SOM. The criterion of visualization quality is the projection error of Sammon’s mapping. Two combinations were considered: (1) a consecutive application of the SOM and Sammon’s mapping and (2) Sammon’s mapping taking into account the learning flow of the self-organizing neural network (integrated combination of the mapping methods). The goal is to obtain a lower projection error and its lower dependence on the so-called “magic factor” in Sammon’s mapping. Different modifications of Sammon’s mapping are examined experimentally and applied in the combination with the SOM. A parallel algorithm of the integrated combination has been proposed.     

Effect of self-constructed concept map with peer feedback on chemistry learning outcome

Concept mapping remains widely used in education. However, little is known about how self-constructed concept maps and peer feedback can improve student learning outcomes in chemistry. We investigated the effects of peer feedback on concept mapping and how it improves students' learning performance in a large second-semester, introductory chemistry course. Three hundred and twenty students were randomly assigned to one of two concept mapping conditions: self-constructed concept map with peer feedback and self-constructed concept map without peer feedback. Each group constructed concept maps that depicted the relationship between concepts on the topic of intermolecular force. The results showed that students in the self-constructed concept map with peer feedback condition outperformed students in the no peer feedback condition in chemistry learning outcome. Overall, this study demonstrates that peer feedback enhances the effectiveness of learning with generative concept maps. The implications and future directions are discussed.     

Gradient modeling for multivariate quantitative data

We propose a new parametric model for continuous data, a “g-model”, on the basis of gradient maps of convex functions. It is known that any multivariate probability density on the Euclidean space is uniquely transformed to any other density by using the gradient map of a convex function. Therefore the statistical modeling for quantitative data is equivalent to design of the gradient maps. The explicit expression for the gradient map enables us the exact sampling from the corresponding probability distribution. We define the g-model as a convex subset of the space of all gradient maps. It is shown that the g-model has many desirable properties such as the concavity of the log-likelihood function. An application to detect the three-dimensional interaction of data is investigated.     

Pseudo-holomorphic maps and bubble trees

This paper proves a strong convergence theorem for sequences of pseudo-holomorphic maps from a Riemann surface to a symplectic manifoldN with tamed almost complex structure. (These are the objects used by Gromov to define his symplectic invariants.) The paper begins by developing some analytic facts about such maps, including a simple new isoperimetric inequality and a new removable singularity theorem. The main technique is a general procedure for renormalizing sequences of maps to obtain “bubbles on bubbles.” This is a significant step beyond the standard renormalization procedure of Sacks and Uhlenbeck. The renormalized maps give rise to a sequence of maps from a “bubble tree”—a map from a wedge Σ V S² V S² V ... →N. The main result is that the images of these renormalized maps converge in L^1,2 ∪C° to the image of a limiting pseudo-holomorphic map from the bubble tree. This implies several important properties of the bubble tree. In particular, the images of consecutive bubbles in the bubble tree intersect, and if a sequence of maps represents a homology class then the limiting map represents this class.     

On the Convergence of Coderivative of SAA Solution Mapping for a Parametric Stochastic Generalized Equation

The aim of this paper is to investigate the convergence properties for Mordukhovich’s coderivative of the solution map of the sample average approximation (SAA) problem for a parametric stochastic generalized equation. It is demonstrated that, under suitable conditions, both the cosmic deviation and the ρ-deviation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic generalized equation converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential convergence rate of coderivatives of the solution maps to the SAA parametric generalized equations is established. The results are used to develop sufficient conditions for the consistency of the Lipschitz-like property of the solution map of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic mathematical program with complementarity constraints.     

Mapping cones and separable states

We study mapping cones and their dual cones of positive maps of the \(n\times n\) matrices into itself. For a natural class of cones there is a close relationship between maps in the cone, super-positive maps, and separable states. In particular the composition of a map from the cone with a map in the dual cone is super-positive, and so the natural state it defines is separable.     

Intrinsic Semiharmonic Maps

For maps from a domain Ω⊂ℝ ^m into a Riemannian manifold N, a functional coming from the norm of a fractional Sobolev space has recently been studied by Da Lio and Rivière. An intrinsically defined functional with a similar behavior also exists, and its first variation can be identified with a Dirichlet-to-Neumann map belonging to the harmonic map problem. The critical points have regularity properties analogous to harmonic maps.     

Bernoulli maps of the interval

The ergodic properties of expanding piecewiseC ² maps of the interval are studied. It is shown that such a map is Bernoulli if it is weak-mixing. Conditions are given that imply weak-mixing (and hence Bernoulliness). Partially supported by NSF Grant MCS74-19388 and the Sloan Foundation.     

Proper maps which are Lipschitz α up to the boundary

In this paper we shall construct proper holomorphic mappings from strictly pseudoconvex domains in Cⁿ into the unit ball in C^N which satisfy some regularity conditions up to the boundary. If we only require continuity of the map, but not more, then there is a large class of such maps (see [2], [3], and [5]). On the other hand, if F is C^k on the closure, k > N ? n + 1, then there is a very small class of such maps. In fact such F must be holomorphic across the boundary (see [1] and [4]). We are interested in maps F that are less than C^N ? n + 1, but more than continuous on the closure. Namely, we want to find out if this is a very small or a large class. Our main result is as follows. Theorem, (a) Let ga < 1/6; then there exists an N = N(α, n) such that we can find a map F: Bⁿ → B^N that is proper, holomorphic, and Lipschitz α up to the boundary, but F is not holomorphic across the boundary. (b) If D is a general strictly pseudoconvex domain with C^∞ -boundary in Cⁿ, then we can find a map F: D → B^N, N = N(α, n), that is proper, holomorphic, and Lipschitz α up to the boundary of D. To do part (a) of the theorem we only need to show that we can find a proper holomorphic map F = (f₁, …, F_N): Bⁿ → B^N that is Lipschitz α and f_N(z) = c(1 - Z₁)^1/6 for some constant c > 0. With this we can in fact ensure that the map in (a) is at most Lipschitz 1/6 on the closure of Bⁿ.     

On the Convergence of Coderivative of SAA Solution Mapping for a Parametric Stochastic Variational Inequality

The aim of this paper is to investigate the convergence properties for Mordukhovich’s coderivative of the solution map of the sample average approximation (SAA) problem for a parametric stochastic variational inequality with equality and inequality constraints. The notion of integrated deviation is introduced to characterize the outer limit of a sequence of sets. It is demonstrated that, under suitable conditions, both the cosmic deviation and the integrated deviation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic variational inequality converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential convergence rate of coderivatives of the solution maps to the SAA parametric stochastic variational inequality is established. The results are used to develop sufficient conditions for the consistency of the Lipschitz-like property of the solution map of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic bilevel program.     

Optimal decisions in combining the SOM with nonlinear projection methods

Visual data mining is an efficient way to involve human in search for a optimal decision. This paper focuses on the optimization of the visual presentation of multidimensional data.A variety of methods for projection of multidimensional data on the plane have been developed. At present, a tendency of their joint use is observed. In this paper, two consequent combinations of the self-organizing map (SOM) with two other well-known nonlinear projection methods are examined theoretically and experimentally. These two methods are: Sammon’s mapping and multidimensional scaling (MDS). The investigations showed that the combinations (SOM_Sammon and SOM_MDS) have a similar efficiency. This grounds the possibility of application of the MDS with the SOM, because up to now in most researches SOM is applied together with Sammon’s mapping. The problems on the quality and accuracy of such combined visualization are discussed. Three criteria of different nature are selected for evaluation the efficiency of the combined mapping. The joint use of these criteria allows us to choose the best visualization result from some possible ones.Several different initialization ways for nonlinear mapping are examined, and a new one is suggested. A new approach to the SOM visualization is suggested.The obtained results allow us to make better decisions in optimizing the data visualization.     

Examples of attracting sets of birkhoff type

We discuss an explicit example of a map of the plane R ² with a nontrivial attracting set. In particular, we are concerned with the concept of rotation number introduced by Poincaré for maps of the circle and its subsequent extension by Birkhoff to maps of the annulus. The use of rotation number allows nontrivial attractors to be distinguished. The map we discuss has an attracting set containing a set of orbits with infinitely many different rotation numbers. We obtain the map by considering an Euler iteration of a family of vector fields originally described by Arnold and find that the resulting Euler map undergoes some bifurcations which are analogous to those of the family of vector fields. Specifically, there are Hopf bifurcations where changes of stability of a fixed point result in the creation of an attracting circle. The circle which grows from the fixed point is then shown to undergo structural changes giving nontrivial attracting sets. This arises from Euler map behaviour for which the corresponding vector field behaviour is a heteroclinic saddle connection. It is possible to give an explicit trapping region for the Euler map which contains the attracting set and to describe some of its properties. Finally, an analogy is drawn with attracting sets which arise for forced oscillators.     

Self‐dual and self‐petrie‐dual regular maps

Regular maps are cellular decompositions of surfaces with the “highest level of symmetry”, not necessarily orientation‐preserving. Such maps can be identified with three‐generator presentations of groups G of the form G = 〈a, b, c|a² = b² = c² = (ab)^k = (bc)^m = (ca)² = … = 1〉; the positive integers k and m are the face length and the vertex degree of the map. A regular map (G;a, b, c) is self‐dual if the assignment b?b, c?a and a?c extends to an automorphism of G, and self‐Petrie‐dual if G admits an automorphism fixing b and c and interchanging a with ca. In this note we show that for infinitely many numbers k there exist finite, self‐dual and self‐Petrie‐dual regular maps of vertex degree and face length equal to k. We also prove that no such map with odd vertex degree is a normal Cayley map. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69:152‐159, 2012     

On non-isotropic harmonic maps of surfaces into complex projective spaces

It is proved by purely algebraic method that weakly conformai, conformai andA _z ³ = 0 are mutually equivalent if ϕ :Ω→ℂP ⁿ is a non-isotropic harmonic map and the harmonic maps with isotropy order ≥3 are uniquely determined by a system of ordinary differential equations. A method is given, by which the isotropy orders of non-isotropic harmonic maps can be computed.     

A generalization of Hahn-Banach extension theorem

In this paper, the notion of affinelike set-valued maps is introduced and some properties of these maps are presented. Then a new Hahn-Banach extension theorem with a K-convex set-valued map dominated by an affinelike set-valued map is obtained.     

Evaluating Kohonen’s learning rule: An approach through genetic algorithms

This paper examines the technical foundations of the self-organising map (SOM). It compares Kohonen’s heuristic-based training algorithm with direct optimisation of a locally-weighted distortion index, also used by Kohonen. Direct optimisation is achieved through a genetic algorithm (GA). Although GAs have been used before with the SOM, this has not been done in conjunction with the distortion index. Comparing heuristic-based training and direct optimisation for the SOM is analogous to comparing the Backpropagation algorithm for feedforward networks with direct optimisation of RMS error. Our experiments reveal lower values of the distortion index with direct optimisation. As to whether the heuristic-based algorithm is able to provide an approximation to gradient descent, our results suggest the answer should be in the negative. Theorems for one-dimensional and for square maps indicate that different point densities will emerge for the two training approaches. Our findings are in accordance with these results.     

Directed complete poset congruences

The study of algebraic properties of ordered structures has shown that their behavior in many cases is different from algebraic structures. For example, the analogues of the fundamental mapping theorem for sets which characterizes surjective maps as quotient sets modulo their kernel relations, is not true for order-preserving maps between posets (partially ordered sets). The main objective of this paper is to study the quotients of dcpos (directed complete partially ordered sets), and their relations with surjective dcpo maps (directed join preserving maps). The motivation of studying such infinitary ordered structures is their importance in domain theory, a theory on the borderline of mathematics and theoretical computer science.In this paper, introducing the notion of a pre-congruence on dcpos (directed complete partially ordered sets), we give a characterization of dcpo congruences. Also, it is proved that unlike natural dcpo congruences, the dcpo congruences are precisely kernels of surjective dcpo maps. Also, while it is known that the image of a dcpo map is not necessarily a subdcpo of its codomain, we find equivalent conditions on a dcpo map to satisfy this property. Moreover, we prove the Decomposition Theorem and its consequences for dcpo maps.     

Semiconjugates of One-dimensional Maps

R -semiconjugate maps are defined as a natural means of relating a map F of R m to a mapping { of the interval via a link map H . Invariants are seen to be special types of semiconjugate links where { is the identity. Basic relationships between the dynamical behaviors of { and F are established, and conditions under which a link map H is a Liapunov function are obtained. Examples and applications involving concepts from stability to chaos are discussed.     

Weak Graph Map Homotopy and Its Applications∗

The authors introduce a notion of a weak graph map homotopy (they call it M-homotopy), discuss its properties and applications. They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001. The difference between them is that the weak graph map homotopy transformation is defined in terms of maps, while the graph homotopy transformation is defined by means of combinatorial operations. They discuss its advantages over the graph homotopy transformation. As its applications, they investigate the mapping class group of a graph and the 1-order MP-homotopy group of a pointed simple graph. Moreover, they show that the 1-order MP-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.     

The principal part of a block map

A block map is a map f{0, 1}ⁿ {0, 1} for some n ? 1. Block maps can be represented by polynomials with coefficients in Z₂. The notion of the principal part of a block map is introduced. It is used to obtain some conditions under which block maps which are linear in the first variable but not linear in all variables are irreducible with respect to composition.