Mediating Mathematical Meaning Through Symbolisation, Verbalisation and Visualisation

2008 ◽  
pp. 111-160
Author(s):  
Elena Nardi
Keyword(s):  
Mathematical Meaning

Rosen-Chambers Variation Theory of Linearly-Damped Classic and Quantum Oscillator

2014 ◽  
Vol 4 (1) ◽  
pp. 404-426
Author(s):  
Vincze Gy. Szasz A.
Keyword(s):  
Harmonic Oscillator ◽  
Classical Mechanics ◽  
Universal Time ◽  
Variation Theory ◽  
Quantum Oscillator ◽  
Variation Principle ◽  
Poisson Brackets ◽  
Mathematical Meaning ◽  
Damped Harmonic Oscillator ◽  
Damped Oscillator

Phenomena of damped harmonic oscillator is important in the description of the elementary dissipative processes of linear responses in our physical world. Its classical description is clear and understood, however it is not so in the quantum physics, where it also has a basic role. Starting from the Rosen-Chambers restricted variation principle a Hamilton like variation approach to the damped harmonic oscillator will be given. The usual formalisms of classical mechanics, as Lagrangian, Hamiltonian, Poisson brackets, will be covered too. We shall introduce two Poisson brackets. The first one has only mathematical meaning and for the second, the so-called constitutive Poisson brackets, a physical interpretation will be presented. We shall show that only the fundamental constitutive Poisson brackets are not invariant throughout the motion of the damped oscillator, but these show a kind of universal time dependence in the universal time scale of the damped oscillator. The quantum mechanical Poisson brackets and commutation relations belonging to these fundamental time dependent classical brackets will be described. Our objective in this work is giving clearer view to the challenge of the dissipative quantum oscillator.

scholarly journals Hvordan kommer matematisk meningsskaping til syne i barns lek? En casestudie

2010 ◽  
Vol 3 (3) ◽  
Author(s):  
Vigdis Flottorp
Keyword(s):  
Day Care ◽  
Meaning Making ◽  
Communicative Competence ◽  
Field Work ◽  
Care Centre ◽  
Mathematical Meaning ◽  
Children's Play ◽  
Physical Context ◽  
Children’S Play ◽  
Day Care Centre

Title: Mathematical meaning making in children’s play? Verbal and non-verbal forms of expressionsAbstract: I analyze an episode from field work in a multilingual day-care centre in Oslo. I examine verbal and non-verbal expressions. The children are 5 years old, and the mathematics is about classification. The children are creating structure and are seeking meaning. This is a key part of their play. My findings indicate that mathematical order and structure become conscious experiences to the children. I argue that we cannot know about the children’s mathematical and communicative competence without knowing the physical context, the play in the sandpit, and the friendship between the boys.

scholarly journals Levels of Sophistication in Elementary Students’ Understanding of Polygon Concept and Polygons Classes

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1966
Author(s):  
Melania Bernabeu ◽  
Salvador Llinares ◽  
Mar Moreno
Keyword(s):  
Elementary Students ◽  
Third Grade ◽  
Mathematical Language ◽  
Mental Process ◽  
Mathematical Meaning ◽  
Mental Operation ◽  
Spatial Structuring ◽  
Four Levels ◽  
Children’S Thinking

This paper reports sophistication levels in third grade children’s understanding of polygon concept and polygon classes. We consider how children endow mathematical meaning to parts of figures and reason to identify relationships between polygons. We describe four levels of sophistication in children’s thinking as they consider a figure as an example of a polygon class through spatial structuring (the mental operation of building an organization for a set of figures). These levels are: (i) partial structuring of polygon concept; (ii) global structuring of polygon concept; (iii) partial structuring of polygon classes; and (iv) global structuring of polygon classes. These levels detail how cognitive apprehensions, dimensional deconstruction, and the use of mathematical language intervene in the mental process of spatial structuring in the understanding of the classes of polygons.

scholarly journals A Unified Scheme for Generalized Sectors Based on Selection Criteria: Order Parameters of Symmetries and of Thermality and Physical Meanings of Adjunctions

2003 ◽  
Vol 10 (03) ◽  
pp. 235-279 ◽  
Author(s):  
Izumi Ojima
Keyword(s):  
Selection Criteria ◽  
Quantum Physics ◽  
Relativistic Quantum ◽  
Order Parameters ◽  
Mathematical Framework ◽  
Mathematical Meaning ◽  
Classifying Space ◽  
Specific Domain ◽  
Geometrical Meaning ◽  
Field Algebra

A unified scheme for treating generalized superselection sectors is proposed on the basis of the notion of selection criteria to characterize states of relevance to each specific domain in quantum physics, ranging from the relativistic quantum fields in the vacuum situations with unbroken and spontaneously broken internal symmetries, through equilibrium and non-equilibrium states to some basic aspects in measurement processes. This is achieved by the help of c → q and q → c channels: the former determines the states to be selected and to be parameterized by the order parameters, and the latter provides the physical interpretations of selected states in terms of order parameters. This formulation extends the traditional range of applicability of the Doplicher-Roberts construction method for recovering the field algebra and the gauge group (of the first kind) from the data of group invariant observables to the situations with spontaneous symmetry breakdown: in use of the machinery proposed, the physical and mathematical meaning of basic structural ingredients associated with the spontaneously broken symmetry are re-examined, such as the degenerate vacua parameterized by the variables belonging to the relevant homogeneous space, the Goldstone modes and condensates, etc. The geometrical meaning of the space of order parameters is naturally understood in relation with the adjunction as the classifying space of a sector structure. As further examples of applications, some basic notions arising in the mathematical framework of quantum theory are reformulated and examined in connection with control theory.

A Mathematical Comparison of the Lachance - Traill Matrix Correction Procedure with Statistical Multiple Linear Regression Analysis in XRF Applications

1992 ◽  
Vol 36 ◽  
pp. 1-10
Author(s):  
Anthony J. Klimasara
Keyword(s):  
Linear Regression ◽  
Regression Model ◽  
Multiple Linear Regression ◽  
Linear Regression Model ◽  
Linear Regression Analysis ◽  
Multiple Linear Regression Model ◽  
Mathematical Meaning ◽  
Matrix Correction ◽  
Independent Variables ◽  
Xrf Analysis

AbstractIt will be shown that the Lachance-Traill XRF matrix correction equations can be derived from the statistical multiple linear regression model. By selecting and properly transforming the independent variables and then applying the statistical multiple linear regression model, the following form of the matrix correction equation is obtained:Furthermore, it will be shown that the Lachance-Traill influence coefficients have a deeper mathematical meaning. They can be related to the multiple regression coefficients of the transformed system:Finally, it will be proposed that the Lachance-Traill model is equivalent to the statistical multiple linear regression model with the transformed independent variables. Knowing these facts will simplify correction subroutines in Quantitative/Empirical XRF Analysis programs. These mathematical facts have already been implemented and presented in a paper: “Automated Quantitative XRF Analysis Software in Quality Control Applications” (Pacific-International Congress on X-ray Analytical Methods, Hawaii, 1991).This demonstrates that the Lachance-Traill model has a strong mathematical foundation and is naturally justified mathematically.

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