**386**

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**Assessing a teacher’s competence for implementation of inquiry based mathematics education from a discussion of open geometrical situations**

Roubíček, Filip

2017 - English
The study is focused on the possibility to assess and develop primary teachers’ abilities to implement some principles of inquiry based mathematics education into their teaching practice. The requirements on a primary teacher and their competence are assessed through a joint discussion of open geometrical situations presented through Concept Cartoons. Teachers’ written comments and notes and verbal statements expressed in the discussion of four geometrical problems were analysed and presented.
Keywords:
*inguiry based mathematics education; primary teacher; elementary geometry*
Available at various institutes of the ASCR
Assessing a teacher’s competence for implementation of inquiry based mathematics education from a discussion of open geometrical situations

The study is focused on the possibility to assess and develop primary teachers’ abilities to implement some principles of inquiry based mathematics education into their teaching practice. The ...

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**Sine-Gordon model in the homogeneous higher grading**

Zuevsky, Alexander

2017 - English
A construction of equations and solutions for the sine-Gordon model in the homogeneous grading as an example of higher grading a ne Toda models are considered.
Keywords:
*soliton solutions; affine Toda modes; homogeneous sine-Gordon model*
Available on request at various institutes of the ASCR
Sine-Gordon model in the homogeneous higher grading

A construction of equations and solutions for the sine-Gordon model in the homogeneous grading as an example of higher grading a ne Toda models are considered.

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**Courant algebroid connections and string effective actions**

Jurčo, B.; Vysoký, Jan

2017 - English
Courant algebroids are a natural generalization of quadratic Lie algebras, appearing in various contexts in mathematical physics. A connection on a Courant algebroid gives an analogue of a covariant derivative compatible with a given fiber-wise metric. Imposing further conditions resembling standard Levi-Civita connections, one obtains a class of connections whose curvature tensor in certain cases gives a new geometrical description of equations of motion of low energy effective action of string theory. Two examples are given. One is the so called symplectic gravity, the second one is an application to the the so called heterotic reduction. All necessary definitions, propositions and theorems are given in a detailed and self-contained way.
Keywords:
*Courant algebroid connections; low-energy effective actions; Levi-Civita connections*
Available on request at various institutes of the ASCR
Courant algebroid connections and string effective actions

Courant algebroids are a natural generalization of quadratic Lie algebras, appearing in various contexts in mathematical physics. A connection on a Courant algebroid gives an analogue of a covariant ...

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**Genus two prime form formula for vertex operator characters**

Zuevsky, Alexander

2017 - English
We find an expression for the self-sewn genus two Riemann surface counterpart of the torus formula for for an $n$-vertex operator character for the Heisenberg vertex operator algebra an $n$-point correlation function for a vertex operator algebra module with complex parameterization of corresponding states.
Keywords:
*prime forms; vertex operator algebras; correlation functions; prime form identities*
Available at various institutes of the ASCR
Genus two prime form formula for vertex operator characters

We find an expression for the self-sewn genus two Riemann surface counterpart of the torus formula for for an $n$-vertex operator character for the Heisenberg vertex operator algebra an $n$-point ...

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**On the motion of chemically reacting fluids through porous medium**

Feireisl, Eduard; Mikyška, J.; Petzeltová, Hana; Takáč, P.

2017 - English
We consider a parabolic-hyperbolic system of nonlinear partial differential equations modeling the motion of a chemically reacting mixture through porous medium. The existence of classical as well as weak solutions is established under several physically relevant choices of the constitutive equations and relevant boundary conditions.
Keywords:
*chemically reacting fluid; porous medium; DiPerna Lions theory*
Available on request at various institutes of the ASCR
On the motion of chemically reacting fluids through porous medium

We consider a parabolic-hyperbolic system of nonlinear partial differential equations modeling the motion of a chemically reacting mixture through porous medium. The existence of classical as well as ...

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**VSI electromagnetic fields**

Ortaggio, Marcello; Pravda, Vojtěch

2017 - English
A p-form F is VSI (i.e., all its scalar invariants of arbitrary order vanish) in a n-dimensional spacetime if and only if it is of type N, its multiple null direction is "degenerate Kundt", and ...F = 0. This recent result is reviewed in the present contribution and its main consequences are summarized. In particular, a subset of VSI Maxwell fields possesses a universal property, i.e., they also solve (virtually) any generalized (non-linear and with higher derivatives) electrodynamics, possibly also coupled to Einstein's gravity.
Keywords:
*p-form electromagnetism; VSI fields*
Available on request at various institutes of the ASCR
VSI electromagnetic fields

A p-form F is VSI (i.e., all its scalar invariants of arbitrary order vanish) in a n-dimensional spacetime if and only if it is of type N, its multiple null direction is "degenerate Kundt", and ...F = ...

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**Mathematical Thermodynamics of Viscous Fluids**

Feireisl, Eduard

2017 - English
This course is a short introduction to the mathematical theory of the motion of viscous fluids. We introduce the concept of weak solution to the Navier-Stokes-Fourier system and discuss its basic properties. In particular, we construct the weak solutions as a suitable limit of a mixed numerical scheme based on a combination of the finite volume and finite elements method. The question of stability and robustness of various classes of solutions is addressed with the help of the relative (modulated) energy functional. Related results concerning weak-strong uniqueness and conditional regularity of weak solutions are presented. Finally, we discuss the asymptotic limit when viscosity of the fluid tends to zero. Several examples of ill- posedness for the limit Euler system are given and an admissibility criterion based on the viscous approximation is proposed.
Keywords:
*thermodynamics of viscous fluid; Navier-Stokes-Fourier system*
Available on request at various institutes of the ASCR
Mathematical Thermodynamics of Viscous Fluids

This course is a short introduction to the mathematical theory of the motion of viscous fluids. We introduce the concept of weak solution to the Navier-Stokes-Fourier system and discuss its basic ...

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**Observing how future primary school teachers reason and generalize: the case of number triangles and Concept Cartoons**

Samková, L.; Tichá, Marie

2017 - English
The contribution focuses on the possibility to use an educational tool called Concept Cartoons as a diagnostic instrument in problem solving and problem posing activities of future primary school teachers. The aim of the presented study is to observe which aspects of future primary school teachers' knowledge related to reasoning and generalization could be investigated through Concept Cartoons that are based on a substantial learning environment called "Number triangles".
Keywords:
*future primary school teachers; reasoning; generalization*
Available at various institutes of the ASCR
Observing how future primary school teachers reason and generalize: the case of number triangles and Concept Cartoons

The contribution focuses on the possibility to use an educational tool called Concept Cartoons as a diagnostic instrument in problem solving and problem posing activities of future primary school ...

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**Average contra-rotation and co-rotation of line segments for flow field analysis**

Šístek, Jakub; Kolář, Václav

2017 - English
The earlier concept of the average co-rotation of infinitesimal radial line segments near a point is extended to the case of contra-rotation. The tensor of the contra-rotation is introduced and averaged over "all planar cross sections" going through the examined point. Both the average contra-rotation and co-rotation, representing shear-free quantities, are applied to describe a complex flow structure.
Keywords:
*fluid mechanics; contra-rotation*
Available on request at various institutes of the ASCR
Average contra-rotation and co-rotation of line segments for flow field analysis

The earlier concept of the average co-rotation of infinitesimal radial line segments near a point is extended to the case of contra-rotation. The tensor of the contra-rotation is introduced and ...

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**Crandall-Rabinowitz type bifurcation for non-differentiable perturbations of smooth mappings**

Recke, L.; Väth, Martin; Kučera, Milan; Navrátil, J.

2017 - English
We consider abstract equations of the type ..., where lambda is a bifurcation parameter and tau is a perturbation parameter. We suppose that ... for all lambda and tau, F is smooth and the unperturbed equation ... describes a Crandall-Rabinowitz bifurcation in lambda=0, that is, two half-branches of nontrivial solutions bifurcate from the trivial solution in lambda=0. Concerning G, we suppose only a certain Lipschitz condition; in particular, G is allowed to be non-differentiable. We show that for fixed small ... there exist also two half-branches of nontrivial solutions to the perturbed equation, but they bifurcate from the trivial solution in two bifurcation points, which are different, in general. Moreover, we determine the bifurcation directions of those two half-branches, and we describe, asymptotically as ..., how the bifurcation points depend on tau. Finally, we present applications to boundary value problems for quasilinear elliptic equations and...
Keywords:
*nonsmooth equation; Lipschitz bifurcation branch; formula for the bifurcation direction*
Available on request at various institutes of the ASCR
Crandall-Rabinowitz type bifurcation for non-differentiable perturbations of smooth mappings

We consider abstract equations of the type ..., where lambda is a bifurcation parameter and tau is a perturbation parameter. We suppose that ... for all lambda and tau, F is smooth and the unperturbed ...

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