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A current research problem in mathematics education is the characterization of the role of teachers in the processes of technology integration in mathematics classrooms. This article shows how two ...

In this paper, we study how undergraduate students define 3D geometrical solids. With this aim, we have identified the routines that are present in the discourse of the students when describing and defining these solids. These routines are one of the properties that characterise the mathematical discourse in the theory of commognition (Sfard 2008). Our results show three different types of routines. The first type is related to the process of describing the solids, the second one to the process of defining the solids and the rest of the routines have a transversal nature. All of them together give us a global vision of the mathematical practice of defining of these undergraduate students. For instance, it seems that some of these students do not have a clear idea of what a definition is. Moreover, there are also differences between the discourse of students when defining 2D figures and the discourse of students when defining 3D solids.

A purely algebraic formulation (i.e., polynomial equations only) of the minimum-cost multi-impulse orbit-transfer problem without time constraints is presented, while keeping all the variables with a precise physical meaning. General algebraic techniques are applied to solve these equations (resultants, Grobner bases, etc.) in several situations of practical interest of different degrees of generality. For instance, a proof of the optimality of the Hohmann transfer for the minimum-fuel two-impulse circular-to-circular orbit-transfer problem is provided. Finally, a general formula is also provided for the optimal two-impulse in-plane transfer between two rotated elliptical orbits under a mild symmetry assumption on the two points where the impulses are applied (which, it is conjectured, can be removed).

Undergraduate students’ engagement with mathematical discourse when defining geometric solids is analysed and compared with what their lecturers expect them to do. The theory of commognition is adopted as the theoretical framework, which permits the characterisation and comparison of their discursive activities, and may lead to the identification of potential commognitive conflicts. The participants were forty-five undergraduate students (primary pre-service teachers) and their lecturers. A worksheet with questions about defining geometric solids was used as a data collection instrument. The students, in small groups, had to discuss and write their answers, and the lecturers were asked what they expected from their students. Results show three main areas of mismatch between students’ engagement in mathematical discourse and what their lecturers expected from them. There is no clear consensus across the students on how to define or on what a definition is or on which criterion to use when selecting a definition.

We provide an approximate zero $$\widetilde{S}(g,L)$$ for the hyperbolic Kepler’s equation $$S-g\,{{\mathrm{arcsinh}}}(S)-L=0$$ for $$g\in (0,1)$$ and $$L\in [0,\infty )$$ . We prove, by using Smale’s $$\alpha $$ -theory, that Newton’s method starting at our approximate zero produces a sequence that converges to the actual solution S(g, L) at quadratic speed, i.e. if $$S_n$$ is the value obtained after n iterations, then $$|S_n-S|\le 0.5^{2^n-1}|\widetilde{S}-S|$$ . The approximate zero $$\widetilde{S}(g,L)$$ is a piecewise-defined function involving several linear expressions and one with cubic and square roots. In bounded regions of $$(0,1) \times [0,\infty )$$ that exclude a small neighborhood of $$g=1, L=0$$ , we also provide a method to construct simpler starters involving only constants.

We study the Riemann curvature tensor of (\kappa,\mu,\nu)-contact metric manifolds, which we prove to be completely determined in dimension 3, and we observe how it is affected by D_a-homothetic deformations. This prompts the definition and study of generalized (\kappa,\mu,\nu)-space forms and of the necessary and sufficient conditions for them to be conformally flat.

We prove that every contact metric ( κ , μ ) -space admits a canonical η -Einstein Sasakian or η -Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of κ and μ for which such metrics are Sasaki–Einstein and paraSasaki–Einstein. Conversely, we prove that, under some natural assumptions, a K-contact or K-paracontact manifold foliated by two mutually orthogonal, totally geodesic Legendre foliations admits a contact metric ( κ , μ ) -structure. Furthermore, we apply the above results to the geometry of tangent sphere bundles and we discuss some geometric properties of ( κ , μ ) -spaces related to the existence of Einstein–Weyl and Lorentzian–Sasaki–Einstein structures.

We study the Riemann curvature tensor of (κ, μ, ν)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (κ, μ, ν)-spaces. We present examples or obstruction results of these spaces in all possible cases.

We present classifications of totally geodesic and totally umbilical Legendrian submanifolds of $(\kappa,\mu)$-spaces with Boeckx invariant $I \leq -1$. In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly construct., Comment: 14 pages

Generalized (\kappa ,\mu)-space forms are introduced and studied. We examine in depth the contact metric case and present examples for all possible dimensions. We also analyse the trans-Sasakian case., Comment: 20 pages, several changes have been done in this version

We will prove that normal complex contact metric manifolds that are Bochner flat must have constant holomorphic sectional curvature 4 and be Kahler. If they are also complete and simply connected, they must be isometric to the odd-dimensional complex projective space $${{\mathbb{C}P^{2n+1}}}$$ (4) with the Fubini-Study metric. On the other hand, it is not possible for normal complex contact metric manifolds to be conformally flat.

In this paper we will show that a Lagrangian, Lorentzian surface $M^2_1$ in a complex pseudo space form $\widetilde M^2_1 (4c)$ is pseudo-isotropic if and only if $M$ is minimal. Next we will obtain a complete classification of all Lagrangian, Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isotropic., Comment: 15 pages

We study non-paraSasakian paracontact metric $(\kappa,\mu)$-spaces with $\kappa=-1$ (equivalent to $h^2=0$ but $h\neq0$). These manifolds, which do not have a contact geometry counterpart, will be classified locally in terms of the rank of $h$. We will also give explicit examples of every possible constant rank of $h$., Comment: 12 pages; several corrections have been made in this version

We study paracontact metric $(\kappa,\mu)$-spaces with $\kappa=-1$, equivalent to $h^2=0$ but not $h=0$. In particular, we will give an alternative proof of Theorem 3.2 of [11] and present examples of paracontact metric $(-1,2)$-spaces and $(-1,0)$-spaces of arbitrary dimension with tensor $h$ of every possible constant rank. We will also show explicit examples of paracontact metric $(-1, \mu)$-spaces with tensor $h$ of non-constant rank, which were not known to exist until now., Comment: 9 pages

We obtain an approximate solution $$\tilde{E}=\tilde{E}(e,M)$$ of Kepler’s equation $$E-e\sin (E)=M$$ for any $$e\in [0,1)$$ and $$M\in [0,\pi ]$$ . Our solution is guaranteed, via Smale’s $$\alpha $$ -theory, to converge to the actual solution $$E$$ through Newton’s method at quadratic speed, i.e. the $$n$$ -th iteration produces a value $$E_n$$ such that $$|E_n-E|\le (\frac{1}{2})^{2^n-1}|\tilde{E}-E|$$ . The formula provided for $$\tilde{E}$$ is a piecewise rational function with conditions defined by polynomial inequalities, except for a small region near $$e=1$$ and $$M=0$$ , where a single cubic root is used. We also show that the root operation is unavoidable, by proving that no approximate solution can be computed in the entire region $$[0,1)\times [0,\pi ]$$ if only rational functions are allowed in each branch.

In the last years, a growing number of researchers have studied paracontact metric geometry. We shall present here some of the recently obtained results in this topic.

We study a remarkable class of paracontact metric manifolds which have no contact metric counterpart: the paracontact metric $(-1,\widetilde\mu)$-spaces which are not paraSasakian (i.e. have $\widetilde h\neq0$). We present explicit examples with $\widetilde h$ of every possible constant rank and some with non-constant rank, which were not known to exist until recently., Comment: 5 pages, a short review of previous papers in the field