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The inclusion of alternative and multimodal methods for online interaction and knowledge construction in mathematics teacher education is still an incipient field. In this paper, we present a multiple case study of three blended courses in an elementary mathematics teacher education program at Western University. In these blended courses, the online component included the construction of collaborative mind maps. Through constructivist grounded theory methods, we analyzed teacher candidates' mind maps as (a) final products, and (b) interaction processes. The resulting theory describes how pre-service mathematics teachers interact and construct knowledge while they engage in online collaborative mind mapping. The study provides insights into the ways that collaborative and multimodal technologies affect mathematics teacher education, and in turn, suggests how its implementation may be improved. [ABSTRACT FROM AUTHOR]

Let ( T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote by ω( x, f) and P( f) the ω-limit set of x under f and the set of periodic points of f, respectively. Write Ω( x, f) = { y| there exist a sequence of points x ∈ T and a sequence of positive integers n < n < ··· such that lim x = x and lim $$f^{n_{k}}$$ ( x ) = y}. In this paper, we show that the following statements are equivalent: (1) f is equicontinuous. (2) ω( x, f) = Ω( x, f) for any x ∈ T. (3) ∩ f ( T) = P( f), and ω( x, f) is a periodic orbit for every x ∈ T and map h: x → ω( x, f) ( x ∈ T) is continuous. (4) Ω( x, f) is a periodic orbit for any x ∈ T. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the convergence problem of the iteration scheme x n+1 = λ n+1 y +(1–λ n+1) T n+1 x n for a family of infinitely many nonexpansive mappings T 1, T 2, . . . in a Hilbert space. It is proved that under suitable conditions this iteration scheme converges strongly to the nearest common fixed point of this family of nonexpansive mappings. The results presented in this paper extend and improve some recent results. [ABSTRACT FROM AUTHOR]

Let R be a ring, M be a R-bimodule and m, n be two fixed nonnegative integers with m + n ≠ 0. An additive mapping δ from R into M is called an (m, n)-Jordan derivation if (m + n)δ(A2) = 2mAδ(A) + 2nδ(A)A for every A in R. In this paper, we prove that every (m, n)-Jordan derivation with m ≠ n from a C* -algebra into its Banach bimodule is zero. An additive mapping δ from R into M is called a (m, n)-Jordan derivable mapping at W in R if (m + n)δ(AB + BA) = 2mδ(A)B + 2mδ(B)A + 2nAδ(B) + 2nBδ(A) for each A and B in R with AB = BA = W. We prove that if M is a unital A-bimodule with a left (right) separating set generated algebraically by all idempotents in A, then every (m, n)-Jordan derivable mapping at zero from A into M is identical with zero. We also show that if A and B are two unital algebras, M is a faithful unital (A, B)-bimodule and U=[AMNB] is a generalized matrix algebra, then every (m, n)-Jordan derivable mapping at zero from U into itself is equal to zero. [ABSTRACT FROM AUTHOR]

Let F be a field, and let $\mathbb{G}$ be the standard Borel subgroup of the symplectic group Sp(2 m, F). In this paper, we characterize the bijective maps ϕ: $\mathbb{G}$ → $\mathbb{G}$ satisfying ϕ[ x, y] = [ ϕ( x), ϕ( y)]. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the following implicit iteration scheme recently introduced by Xu and Ori [ Numer. Funct. Anal. Optim., 22, (2001) 767–773]: and to prove several strongly and weakly convergent theorems of the iteration for a finite family of pseudocontractive mappings under condition α n ∈ (0, b] ⊂ (0, 1). [ABSTRACT FROM AUTHOR]

This paper considers the isometric extension problem concerning the mapping from the unit sphere S 1( E) of the normed space E into the unit sphere S 1( l ∞(Γ)). We find a condition under which an isometry from S 1( E) into S 1( l ∞(Γ)) can be linearly and isometrically extended to the whole space. Since l ∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V 0: S 1( E) → S 1( F) is a surjective isometry, where c 00(Γ) ⊆ F ⊆ l ∞(Γ), then V 0 can be extended to be an isometric operator defined on the whole space. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a relaxed extragradient method for finding solutions of a general system of variational inequalities with inverse-strongly monotone mappings in a real Hilbert space. First, this system of variational inequalities is proven to be equivalent to a fixed point problem of nonexpansive mapping. Second, by using the demi-closedness principle for nonexpansive mappings, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient method converges strongly to a solution of this system of variational inequalities. In addition, utilizing this result, we provide some applications of the considered problem not just giving a pure extension of existing mathematical problems. [ABSTRACT FROM AUTHOR]

In 1970, Farris introduced a procedure that can be used to transform a tree metric into an ultra metric. Since its discovery, Farris’ procedure has been used extensively within phylogenetics where it has become commonly known as the Farris transform. Remarkably, the Farris transform has not only been rediscovered several times within phylogenetics, but also in other fields. In this paper, we will review some of its various properties and uses. The paper is divided into four parts and, altogether, 12 sections. In the first part, we introduce a standardized scheme for classifying those dissimilarity mappings to which the Farris transform can be applied—a scheme that has evolved over the years, but has apparently not been spelled out before in sufficient detail. In the second part, we will discuss how a straightforward generalization of the Farris transform naturally arises in T-Theory. The third part describes how this generalized Farris transform can be used to approximate dissimilarities by tree metrics. And in the final part, we describe some further, “non-standard” applications of the Farris transform. [ABSTRACT FROM AUTHOR]

An open problem on orthogonality spaces posed by Jürg Rätz in [10] (also cf. [14] is completely solved in this paper, so that orthogonality spaces admitting nonzero even orthogonally additive mappings are completely described. As a by-product, a characterization of real inner product spaces is also given. [ABSTRACT FROM AUTHOR]

Let X be a real Hilbert space with dim X ≥ 2 and let Y be a real normed space which is strictly convex. In this paper, we generalize a theorem of Benz by proving that if a mapping f, from an open convex subset of X into Y, has a contractive distance ρ and an extensive one Nρ (where N ≥ 2 is a fixed integer), then f is an isometry. [ABSTRACT FROM AUTHOR]

In the paper, the spaces of weakly additive τ-smooth and Radon functionals are investigated. It is proved that the functors of weakly additive τ-smooth and Radon functionals weakly preserve the density of Tychonoff spaces, and the functor of weakly additive τ-smooth functionals forms a monad in the category of Tychonoff spaces and their continuous mappings. Examples and remarks are given showing that these functors fail to satisfy certain Shchepin normality conditions. Problems having positive solutions for normal functors are presented. [ABSTRACT FROM AUTHOR]

In this paper, we .rst derive the representation theorem of onto isometric mappings in the unit spheres ofl1(G) type spaces, and then conclude that such mappings can be extended to the whole space as real linear isometries by using a previous result of the author. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of vector quasi-equilibrium problems with set-valued maps. Almost all the vector equilibrium models of the Blum-Oettli type in the literature are special cases of our new class of equilibrium problems under consideration. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the γ-diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for such vector equilibrium problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a concept of graph convergence for the $$H(\cdot ,\cdot )$$ -co-accretive mapping in Banach spaces and prove an equivalence theorem between graph convergence and resolvent operator convergence for the $$H(\cdot ,\cdot )$$ -co-accretive mapping. Further, we consider a system of generalized variational inclusions involving $$H(\cdot ,\cdot )$$ -co-accretive mapping in real $$q$$ -uniformly smooth Banach spaces. Using resolvent operator technique, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions under some suitable conditions. Further, we discuss the convergence of iterative algorithm using the concept of graph convergence. Our results can be viewed as a refinement and generalization of some known results in the literature. [ABSTRACT FROM AUTHOR]

This paper is devoted to the further development of the theory of almost geodesic mappings of spaces with affine connection. [ABSTRACT FROM AUTHOR]

In this paper, by using the parametric representation of spirallike mappings, we construct to obtain the growth theorem of spirallike mappings on Reinhardt domain B. Moreover, the distortion theorem of spirallike mappings is obtained. [ABSTRACT FROM AUTHOR]

In this paper we introduce an iterative algorithm for finding a common element of the fixed point set of an asymptotically strict pseudocontractive mapping S in the intermediate sense and the solution set of the minimization problem (MP) for a convex and continuously Frechet differentiable functional in Hilbert space. The iterative algorithm is based on several well-known methods including the extragradient method, CQ method, Mann-type iterative method and hybrid gradient projection algorithm with regularization. We obtain a strong convergence theorem for three sequences generated by our iterative algorithm. In addition, we also prove a new weak convergence theorem by a modified extragradient method with regularization for the MP and the mapping S. [ABSTRACT FROM AUTHOR]

In this paper we establish relaxation of an arbitrary 1-equivariant wave map from $${\mathbb{R}^{1+3}_{t,x}{\setminus} (\mathbb{R}\times B(0,1))\to S^3}$$ of finite energy and with a Dirichlet condition at r = 1, to the unique stationary harmonic map in its degree class. This settles a recent conjecture of Bizoń, Chmaj, Maliborski who observed this asymptotic behavior numerically. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the problem of existence of nonequivalent definitions of topological transitivity, which is a classical problem of the topological dynamics. In particular, we use the fact that all available definitions of this kind imply a condition imposed on the dynamical system. The main result of our investigations is the complete classification of these dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, a new concept of the common property (E.A) for two hybrid pairs of mappings is introduced in Menger PM-spaces. Utilizing this concept, some common fixed point theorems, which shed some new light on the study of fixed point results for hybrid pairs in Menger PM-spaces, are obtained under strict contractive conditions. The corresponding results in metric spaces which generalize many known results are also obtained. Finally, an example is also given to exemplify our main results. [ABSTRACT FROM AUTHOR]

In this paper we use shear construction to generate certain subclasses of harmonic univalent mappings with directional convexity because it allows us to study such functions by examining their related conformal univalent mappings. In this setting we find growth, distortion, and coefficient bounds for harmonic univalent mappings that are convex in both the directions of real axis and imaginary axis. [ABSTRACT FROM AUTHOR]

A new subclass of multivalent harmonic functions defined by convolution is introduced in this paper. The subclass generates known subclasses of multivalent harmonic functions, and thus provides a unified treatment in the study of these subclasses. Sufficient coefficient conditions are obtained that are also shown to be necessary when the functions have negative coefficients. Growth estimates and extreme points are also determined. In addition conditions for starlikeness of the Dziok-Srivastava linear operator involving the generalized hypergeometric functions are discussed. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the strong convergence of an implicit iteration process with errors to a common fixed point for a finite family of asymptotically pseudocontractive mappings and nonexpansive mappings in normed linear spaces. The results in this paper improve and extend the corresponding results of Xu and Ori, Zhou and Chang, Sun, Yang and Yu in some aspects. [ABSTRACT FROM AUTHOR]

In this paper direct proofs of some common fixed point results for two and three mappings under weak contractive conditions are given. Some of these results are improved by using different arguments of control functions. Examples are presented showing that some generalizations can not be obtained and also that our results are distinct from the existing ones. [ABSTRACT FROM AUTHOR]

In this paper, we study a coupled system of nonlinear fractional differential equations with multi-point boundary conditions. The differential operator is taken in the Riemann-Liouville sense. Applying the Schauder fixed-point theorem and the contraction mapping principle, two existence results are obtained for the following system {D0+αx(t) = f (t, y(t), D0+py(t)), t ∈ (0, 1), {D0+βy(t) = g (t, x(t), D0+qx(t)), t ∈ (0, 1), {x(0) = x'(0) = x''(0) = ⋯ = x(m-2)(0) = 0, x(1) = λx(ξ), {y(0) = y'(0) = y''(0) = ⋯ = y(m-2)(0) = 0, y(1) = λy(ξ), where 0 < ξ < 1, m ∈ N; m ≥ 2, α, β ∈ (m-1, m) and α, β, p, q, λ satisfy certain conditions. [ABSTRACT FROM AUTHOR]

In [8 Chapter 4.3] Kirchheim and Preiss gave an example of a set K consisting of five 2 × 2 symmetric matrices without rank-one connections, for which there exists a Lipschitz mapping u satisfying $${Du \in K}$$ . In the present paper we construct the rank-one convex hull of K. As a corollary we obtain that for each $${F \in {\rm int}\,K^{rc}}$$ there exists a Lipschitz mapping u satisfying Moreover, we show that the rank-one convex hull of K and the quasiconvex hull of K are equal. [ABSTRACT FROM AUTHOR]

Let C be a nonempty closed convex subset of a Banach space E with the dual E*, let T: C→ E* be a Lipschitz continuous mapping and let S: C→ C be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator, we study the following variational inequality (for short, VI( T− f, C)): find x∈ C such that where f∈ E* is a given element. Utilizing the modified Ishikawa iteration and the modified Halpern iteration for relatively nonexpansive mappings, we propose two modified versions of J.L. Li’s (J. Math. Anal. Appl. 295:115–126, ) iterative algorithm for finding approximate solutions of VI( T− f, C). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of VI( T− f, C), which is also a fixed point of S. [ABSTRACT FROM AUTHOR]

It is known that optimal mappings in optimal transportation problems are uniquely determined by corresponding potential functions. In this paper we prove various local properties of potential functions. In particular we obtain the C1, α regularity of potential functions with optimal exponent α, which improves previous regularity results of Loeper. [ABSTRACT FROM AUTHOR]

In this paper, we study Levitin–Polyak type well-posedness for generalized vector variational inequality problems with abstract and functional constraints. Various criteria and characterizations for these types of well-posednesses are given. [ABSTRACT FROM AUTHOR]

We introduce the concept of θ-derivations on JB*-triples and prove the Hyers–Ulam-Rassias stability of θ-derivations on JB*-triples. We deal with the Hyers-Ulam-Rassias stability that was first introduced by Th. M. Rassias in the paper “On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300”. [ABSTRACT FROM AUTHOR]

Recently, Pathak [13] has made an extension of the notion of compatibility to weak compatibility, and extended the coincidence theorem for compatible mappings in Kaneko and Sessa [11] to weakly compatible mappings [13]. In the present paper, we define a new class of weakly compatible mappings (Definition 4) and prove some common fixed point theorems for these mappings, which satisfy Condition (2) below. Although our main theorem is formulated for weakly compatible mappings, its corresponding formulation for commutative mappings is also a new result, thus presenting a generalization of some theorems of Fisher, Das and Naik, Khan and Kubiaczyk, Reich, Ćirić and Rhoades and Watson. [ABSTRACT FROM AUTHOR]

In this paper we deal with oriented rectilinear congruences in a three-dimensional Euclidean space E 3 establishing a conformal mapping between their middle surface and their middle envelope. We give some properties and determine a special class of them, which have a minimal middle envelope. [ABSTRACT FROM AUTHOR]

Let X, Y be two finite–dimensional topological vector spaces, Z a Hausdorff topological vector space, K ⊂ X and D ⊂ Z be two nonempty sets, C be a pointed, closed, and convex cone in Y with int C ≠ Ø. Let S : K → 2 K and T : K → 2 D be two multivalued mappings, and φ : K× D× K → Y be a trifunction. In this paper, we consider the generalized vector quasi–equilibrium problem, which is formulated by finding $\hat x$ ∈ K and ŷ ∈ T( $\hat x$ ) such that $\hat x$ ∈ S( $\hat x$ ) and φ( $\hat x$ , ŷ, u) /∈ −int C for all u ∈ S( $\hat x$ ). We establish an existence result in which T is not supposed to have any continuity property. Our results extend and improve the corresponding results of Cubiotti, Yao and Guo. [ABSTRACT FROM AUTHOR]

This paper attempts to make accessible a body of ideas surrounding the following result: Typical families of (possibly multi-model) 1-dimensional maps passing through ``Misiurewicz points'' have invariant densities for positive measure sets of parameters. [ABSTRACT FROM AUTHOR]

In this paper, two concepts of well-posedness for quasivariational inequalities having a unique solution are introduced. Some equivalent characterizations of these concepts and classes of well-posed quasivariational inequalities are presented. The corresponding concepts of well-posedness in the generalized sense are also investigated for quasivariational inequalities having more than one solution. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The iterative process is based on the so-called extragradient method. We obtain a weak convergence theorem for two sequences generated by this process. [ABSTRACT FROM AUTHOR]

The Chang-Łoś-Suszko theorem of first-order model theory characterizes universal-existential classes of models as just those elementary classes that are closed under unions of chains. This theorem can then be used to equate two model-theoretic closure conditions for elementary classes; namely unions of chains and existential substructures. In the present paper we prove a topological analogue and indicate some applications. [ABSTRACT FROM AUTHOR]

The Gauss-Lucas Theorem on the roots of polynomials nicely simplifies the computation of the subderivative and regular subdifferential of the abscissa mapping on polynomials (the maximum of the real parts of the roots). This paper extends this approach to more general functions of the roots. By combining the Gauss-Lucas methodology with an analysis of the splitting behavior of the roots, we obtain characterizations of the subderivative and regular subdifferential for these functions as well. In particular, we completely characterize the subderivative and regular subdifferential of the radius mapping (the maximum of the moduli of the roots). The abscissa and radius mappings are important for the study of continuous and discrete time linear dynamical systems. [ABSTRACT FROM AUTHOR]

The paper is devoted to studying generalized differential properties of distance functions that play a remarkable role in variational analysis, optimization, and their applications. The main object under consideration is the distance function of two variables in Banach spaces that signifies the distance from a point to a moving set. We derive various relationships between Fréchet-type subgradients and limiting (basic and singular) subgradients of this distance function and corresponding generalized normals to sets and coderivatives of set-valued mappings. These relationships are essentially different depending on whether or not the reference point belongs to the graph of the involved set-valued mapping. Our major results are new even for subdifferentiation of the standard distance function signifying the distance between a point and a fixed set in finite-dimensional spaces. The subdifferential results obtained are applied to deriving efficient dual-space conditions for the local Lipschitz continuity of distance functions generated by set-valued mappings, in particular, by those arising in parametric constrained optimization. [ABSTRACT FROM AUTHOR]

The paper is devoted to the analysis of the calmness property for constraint set mappings. After some general characterizations, specific results are obtained for various types of constraints, e.g., one single nonsmooth inequality, differentiable constraints modeled by polyhedral sets, finitely and infinitely many differentiable inequalities. The obtained conditions enable the detection of calmness in a number of situations, where the standard criteria (via polyhedrality or the Aubin property) do not work. Their application in the framework of generalized differential calculus is explained and illustrated by examples associated with optimization and stability issues in connection with nonlinear complementarity problems or continuity of the value-at-risk. [ABSTRACT FROM AUTHOR]

A few weak and strong convergence theorems of the modified three-step iterative sequence with errors and the modified Ishikawa iterative sequence with errors for asymptotically non-expansive mappings in any non-empty closed convex subsets of uniformly convex Banach spaces are established. The results presented in this paper substantially extend the results due to Chang (2001), Osilike and Aniagbosor (2000), Rhoades (1994) and Schu (1991). [ABSTRACT FROM AUTHOR]

In this paper, we prove the generalized Hyers-Ulam-Rassias stability of universal Jensen’s equations in Banach modules over a unitalC*-algebra. It is applied to show the stability of universal Jensen’s equations in a Hilbert module over a unitalC*-algebra. Moreover, we prove the stability of linear operators in a Hilbert module over a unitalC*-algebra. [ABSTRACT FROM AUTHOR]

In this paper we investigate holomorphically projective mappings of generalized Kählerian spaces. In the case of equitorsion holomorphically projective mappings of generalized Kählerian spaces we obtain five invariant geometric objects for these mappings. [ABSTRACT FROM AUTHOR]

The achromatic index of a graph G is the maximum number of colours that can be assigned to edges of G in such a way that the colouring is proper and any two colours meet on two adjacent edges. To compute the achromatic index of complete graphs seems to be a very difficult task. Indeed, that knowledge would yield all odd projective plane orders. The aim of the paper is to establish a nontrivial lower bound of the achromatic index of for a prime q≥7. [ABSTRACT FROM AUTHOR]

In this paper, we study singular points and singular values of mappings defined on a Hilbert manifold with values in a finite-dimensional manifold. We give a generalization of Sard's theorem and establish some properties of the Lagrangian derivative of a mapping at a singular point. [ABSTRACT FROM AUTHOR]

We prove common fixed point theorems for three commuting self-mappings on complete metric spaces which generalize and unify some previously known results. Examples and applications to Menger spaces and random operator equations are also given. [ABSTRACT FROM AUTHOR]

In this paper we shall deal with a weak version of the Borsuk-Ulam theorem for G-isovariant maps, which we call the weak isovariant Borsuk-Ulam theorem. One of the results is that the weak isovariant Borsuk-Ulam theorem in linear G-spheres holds for an arbitrary compact Lie group G. On the contrary the weak isovariant Borsuk-Ulam theorem in semilinear G-(homology) spheres holds if and only if G is solvable. [ABSTRACT FROM AUTHOR]

Abstract We study the Bloch constant for K-quasiconformal holomorphie mappings of the unit ball B of C[sup n] into C[sup n]. The final result we prove in this paper is: if f is a K-quasiconformal holomorphic mapping of B into C[sup n] such that det(f(0)) = 1, then f(B) contains a schlicht ball of radius at least, [Multiple line equation(s) cannot be represented in ASCII text] where C[sub n] > 1 is a constant depending on n only, and C[sub n] Arrow right Square root of 10 as n Arrow right Infinity Keywords K-quasiformal mapping, Bloch constant [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the topologically weak concepts of topological groupoids by giving the concepts of α-topological groupoid and α-topological subgroupoid. Furthermore, we show the role of the density condition to allow α-topological subgroupoid inherited properties from α-topological groupoid and the irresoluteness property for the structure maps in α-topological groupoid is studied. We also give some results about the fibers of α-topological groupoids. [Copyright &y& Elsevier]