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A particular smooth interpolation that generates splines
Segeth, Karel
2017 - English
There are two grounds the spline theory stems from -- the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We use the general variational approach called $it smooth interpolation$ introduced by Talmi and Gilat and show that it covers not only the cubic spline and its 2D and 3D analogues but also the well known tension spline (called also spline with tension). We present the results of a 1D numerical example that characterize some properties of the tension spline.
Keywords:
data interpolation; smooth interpolation; spline interpolation
Available in digital repository of the ASCR
A particular smooth interpolation that generates splines
There are two grounds the spline theory stems from -- the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one ...
The Cramér-Rao inequality on singular statistical models
Le, Hong-Van; Jost, J.; Schwachhöfer, L.
2017 - English
We introduce the notions of essential tangent space and reduced Fisher metric and extend the classical Cramér-Rao inequality to $2$-integrable (possibly singular) statistical models for general $varphi$-estimators, where $varphi$ is a $V$-valued feature function and $V$ is a topological vector space. We show the existence of a $varphi$-efficient estimator on strictly singular statistical models associated with a finite sample space and on a class of infinite dimensional exponential models that have been discovered by Fukumizu. We conclude that our general Cramér-Rao inequality is optimal.
Keywords:
Cramér-Rao inequality; Singular Statistical Models
Available on request at various institutes of the ASCR
The Cramér-Rao inequality on singular statistical models
We introduce the notions of essential tangent space and reduced Fisher metric and extend the classical Cramér-Rao inequality to $2$-integrable (possibly singular) statistical models for general ...
Note on the problem of dissipative measure-valued solutions to the compressible non-Newtonian system
Al Baba, Hind; Caggio, Matteo; Ducomet, B.; Nečasová, Šárka
2017 - English
We introduce a dissipative measure-valued solution to the compressible non-Newtonian system. We generalized a result given by Novotný, Nečasová [14]. We derive a relative entropy inequality for measure-valued solution as an extension of the classical entropy inequality introduced by Dafermos [2], Mellet-Vasseur [11], Feireisl-Jin-Novotný [5].
Keywords:
dissipative measure-valued solution; non-Newtonian fluid; compressible fluid
Available on request at various institutes of the ASCR
Note on the problem of dissipative measure-valued solutions to the compressible non-Newtonian system
We introduce a dissipative measure-valued solution to the compressible non-Newtonian system. We generalized a result given by Novotný, Nečasová [14]. We derive a relative entropy inequality for ...
Note on the use of Camassa-Holm equations for simulation of incompressible fluid turbulence
Caggio, Matteo; Bodnár, Tomáš
2017 - English
The aim of this short communication is to briefly introduce the Camassa-Holm equations as a working model for simulation of incompressible fluid turbulence. In particular we discuss its application for turbulent boundary layer flows. This model (and related models) is studied for several years in mathematical community, starting from Leray [23]. It can be understood as a generalization of some classical fluid models (Navier-Stokes equations, Prandtl boundary layer equations), showing some interesting mathematical properties in the analysis of the behavior of it's solution (e.g. Layton and Lewandowski [22]). It has been found however, that the model predictions can lead to surprising extensions of the use of the model in technical applications, namely in simulating the turbulent fluid flows. This brief paper should be understood as an introductory note to this novel class of models for applied scientists.
Keywords:
Camassa-Holm equations; turbulent boundary layer
Available on request at various institutes of the ASCR
Note on the use of Camassa-Holm equations for simulation of incompressible fluid turbulence
The aim of this short communication is to briefly introduce the Camassa-Holm equations as a working model for simulation of incompressible fluid turbulence. In particular we discuss its application ...
On the Friedmann equation for the three-dimensional hypersphere
Křížek, Michal; Mészáros, A.
2016 - English
The present standard cosmological model of the evolution of our universe, is based on the Friedmann equation, which was published by Alexander Friedmann in 1922. He applied Einstein’s equations to an expanding threedimensional sphere which enabled him to avoid boundary conditions. However, his description was very brief. Therefore, the main objective of this article is to detailed a derivation of the Friedmann equation for an unknown expansion function a = a(t) representing the radius of the universe. Furthermore, we present serious arguments showing why the validity of Einstein’s equations should not be extrapolated to the entire universe.
Keywords:
Einstein's equations; standard cosmological model; dark matter; dark energy
Available in digital repository of the ASCR
On the Friedmann equation for the three-dimensional hypersphere
The present standard cosmological model of the evolution of our universe, is based on the Friedmann equation, which was published by Alexander Friedmann in 1922. He applied Einstein’s equations to an ...
Developing open approach to mathematics in future primary school teachers
Samková, L.; Tichá, Marie
2016 - English
In our contribution we focus on the possibility to develop open approach to mathematics in future primary school teachers during a university course on mathematics conducted in inquiry-based manner. We analyse data obtained in the beginning and in the end of the course with respect to two main aspects related to open approach to mathematics: searching for all solutions of a task, and acceptance of different forms of notation of a given solution. Data analysis revealed in the participants three different shifts towards open approach to mathematics, and showed that after the active participation in the course each of the participants improved at least in one of the monitored aspects, and that none of the participants got worse in any of the aspects.
Keywords:
argumentaion; concept Cartoons; future primary school teachers; inquiry-based
Fulltext is available at external website.
Developing open approach to mathematics in future primary school teachers
In our contribution we focus on the possibility to develop open approach to mathematics in future primary school teachers during a university course on mathematics conducted in inquiry-based manner. ...
Numerical modeling of magnetostrictive materials
Kholmetska, I.; Chleboun, J.; Krejčí, Pavel
2016 - English
An operator-differential model for magnetostrictive energy harvesting proposed in the literature is used to calculate the amount of harvested energy under some uncertainty in the Preisach density function. The uncertainty is modeled through a fuzzy set approach. In galfenol, an alloy of gallium and iron, however, the hysteresis phenomenon is present only weakly. This allows to propose a simpler, but computationally faster model without hysteresis. The new, simpler model has been identified from the measured magnetic and magnetostrictive cycles. The amount of harvested energy is of the same order in both models. Again, an uncertainty in the identified model can be considered and its impact on the amount of the harvested energy calculated.
Keywords:
magnetostrictive materials; hysteresis; energy harvesting
Available in digital repository of the ASCR
Numerical modeling of magnetostrictive materials
An operator-differential model for magnetostrictive energy harvesting proposed in the literature is used to calculate the amount of harvested energy under some uncertainty in the Preisach density ...
Anthropic principle and the local Hubble expansion
Křížek, Michal; Somer, L.
2016 - English
We claim that a local expansion of the universe, whose rate is comparable with the Hubble constant H0, had an essential influence on the development of intelligent life on the Earth. We present more than 10 examples showing that some antigravitational effects of the cosmological constant are observable locally in the Solar system. It is known that the luminosity of the Sun increased approximately linearly within the last 4.5Gyr starting at 70% of its present value. We give several independent arguments showing that the average Earth-Sun distance increases about 5m/yr due to antigravitational forces and such a large recession speed cannot be explained by solar wind, tidal forces, plasma outbursts from the Sun, or by the decrease of the Solar mass due to nuclear reactions. Models based on Newtonian mechanics can explain only a few cm per year.
Keywords:
antigravity; habitable zone; law of conservation of energy
Available in digital repository of the ASCR
Anthropic principle and the local Hubble expansion
We claim that a local expansion of the universe, whose rate is comparable with the Hubble constant H0, had an essential influence on the development of intelligent life on the Earth. We present more ...
Error patterns in computation in concept cartoons
Samková, L.; Tichá, Marie; Hošpesová, A.
2015 - English
The paper focuses on an educational tool called Concept Cartoons, and its possible usage in primary school mathematics, especially in promoting discussion in the classroom. We show in detail how to prepare Concept Cartoons displaying various mental computation strategies and error patterns.
Keywords:
concept Cartoons; error patterns; mental computation strategies
Available at various institutes of the ASCR
Error patterns in computation in concept cartoons
The paper focuses on an educational tool called Concept Cartoons, and its possible usage in primary school mathematics, especially in promoting discussion in the classroom. We show in detail how to ...
Word problems of a given structure in the perspective of teacher training
Tichá, Marie; Hošpesová, A.
2015 - English
The paper reports on the results of a research study focusing on how to train prospective teachers for guiding their pupils through the process of solving word problems. The teaching experiment involved prospective teachers’ work with graphical representations called branched chains to solve and pose word problems. The paper shows the difficulties the students were facing, presents ideas from discussions during joint reflection, tries to describe the changes in the students’ skills and attitudes. It also points at the differences between prospective primary and secondary mathematics teachers.
Keywords:
inquiry based mathematics education; prospective teaching training; subject matter knowledge; knowledge base for teaching
Fulltext is available at external website.
Word problems of a given structure in the perspective of teacher training
The paper reports on the results of a research study focusing on how to train prospective teachers for guiding their pupils through the process of solving word problems. The teaching experiment ...
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