Search results for "algebra"

Games and Bisimulations for Intuitionistic First-Order Kripke Models

2021

The aim of this paper is to introduce the notion of a game for intuitionisticfirst-order Kripke models. We also establish links between notions presented here and thenotions of logical equivalence and bounded bisimulation for intuitionistic first-order Kripkemodels, and the Ehrenfeucht–Fra ̈ıss ́e game for classical first-order structures.

Witt vectors and Fermat quotients

2008

Abstract We give a representation of any integer as a vector of the Witt ring W ( Z p ) and relate it to the Fermat quotient q ( n ) = ( n p − 1 − 1 ) / p . Logarithms are introduced in order to establish an isomorphism between the commutative unipotent groups 1 + p W ( Z p ) and W ( Z p ) .

Domain Specific Knowledge Representation for an Intelligent Tutoring System to Teach Algebraic Reasoning

2012

Translation of word problems into symbolic notation is one of the most challenging steps in learning the algebraic method. This paper describes a domain-specific knowledge representation mechanism to support Intelligent Tutoring Systems (ITS) which focus on this stage of the problem solving process. The description language proposed is based on the concept of a hypergraph and makes it possible to simultaneously a) represent all potential algebraic solutions to a given word problem; b) keep track of the student's actions; c) provide automatic remediation; and d) unequivocally determine the current state of the resolution process. An experimental evaluation with students at a public school su…

The Fifth Lesson: Students’ Responses to a Patterning Task Across the Four Countries

2019

The chapter reports a comparative analysis of problem-solving activities in classrooms in the four countries. The problem students work with is a patterning task, taken from a major international comparative study. The idea of analyzing this group work across countries is to see what traces of algebra learning may be discerned in the work of the students. The results show the complexities of early algebra learning by presenting in detail the obvious variations in how students tackle the problem in more or less algebraic ways. In the small-group work in classrooms, students pool their ideas allowing for the discourse to build on itself in a cumulative manner.

Gradient design for liquid chromatography using multi-scale optimization.

2017

Abstract In reversed phase-liquid chromatography, the usual solution to the “general elution problem” is the application of gradient elution with programmed changes of organic solvent (or other properties). A correct quantification of chromatographic peaks in liquid chromatography requires well resolved signals in a proper analysis time. When the complexity of the sample is high, the gradient program should be accommodated to the local resolution needs of each analyte. This makes the optimization of such situations rather troublesome, since enhancing the resolution for a given analyte may imply a collateral worsening of the resolution of other analytes. The aim of this work is to design mul…

An atlas- and data-driven approach to initializing reaction-diffusion systems in computer cardiac electrophysiology

2016

The cardiac electrophysiology (EP) problem is governed by a nonlinear anisotropic reaction-diffusion system with a very rapidly varying reaction term associated with the transmembrane cell current. The nonlinearity associated with the cell models requires a stabilization process before any simulation is performed. More importantly, when used in a 3-dimensional (3D) anatomy, it is not sufficient to perform this stabilization on the basis of isolated cells only, since the coupling of the different cells through the tissue greatly modulates the dynamics of the system. Therefore, stabilization of the system must be performed on the entire 3D model. This work develops a novel procedure for the i…

Estimation of peak capacity based on peak simulation.

2018

Peak capacity (PC) is a key concept in chromatographic analysis, nowadays of great importance for characterising complex separations as a criterion to find the most promising conditions. A theoretical expression for PC estimation can be easily deduced in isocratic elution, provided that the column plate count is assumed constant for all analytes. In gradient elution, the complex dependence of peak width with the gradient program implies that an integral equation has to be solved, which is only possible in a limited number of situations. In 2005, Uwe Neue developed a comprehensive theory for the calculation of PC in gradient elution, which is only valid for certain situations: single linear …

2019

Abstract The aim of this work is to present a formulation to solve the one-dimensional Ising model using the elementary technique of mathematical induction. This formulation is physically clear and leads to the same partition function form as the transfer matrix method, which is a common subject in the introductory courses of statistical mechanics. In this way our formulation is a useful tool to complement the traditional more abstract transfer matrix method. The method can be straightforwardly generalised to other short-range chains, coupled chains and is also computationally friendly. These two approaches provide a more complete understanding of the system, and therefore our work can be o…

Ray matrix analysis of anamorphic fractional Fourier systems

2006

In this work we extend the application of the ray matrix approach to analyse anamorphic fractional Fourier systems, i.e., fractional Fourier optical systems where the fractional power is different for two orthogonal directions. The application of the ray matrix approach allows for easily obtaining the properties of the optical system, and it is therefore a powerful tool to design and simplify complicated systems. For simplicity we consider fractional Fourier systems with real orders and systems without apertures. We start by presenting the analysis of some previously reported anamorphic Fourier and fractional Fourier systems, and we end by proposing a simple optical system with tunable anam…

Localized potentials in electrical impedance tomography

2008

In this work we study localized electric potentials that have an arbitrarily high energy on some given subset of a domain and low energy on another. We show that such potentials exist for general L ∞ -conductivities in almost arbitrarily shaped subregions of a domain, as long as these regions are connected to the boundary and a unique continuation principle is satisfied. From this we deduce a simple, but new, theoretical identifiability result for the famous Calderon problem with partial data. We also show how to con- struct such potentials numerically and use a connection with the factorization method to derive a new non-iterative algorithm for the detection of inclusions in electrical imp…