Differentiated Math Instruction: Instruction that Incorporates the Common Core State Standards and Effectively Addresses the Needs of Sixth Grade Students. A Self-Study.

My capstone was a self-study conducted in a sixth grade 15:1 classroom. The purpose of my research was to determine how to incorporate the Common Core State Standards for math and effectively meet the needs of sixth grade students with different levels of abilities. Different strategies and interventions were used in the classroom during one hour class period and then were reflected into a journal to document the experience

A generalization of the Schauder fixed point theorem via multivalued contractions

summary:We establish a fixed point theorem for a continuous function $f:X\to E$, where $E$ is a Banach space and $X\subseteq E$. Our result, which involves multivalued contractions, contains the classical Schauder fixed point theorem as a special case. An application is presented

On the two-point problem for implicit second-order ordinary differential equations

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On the Cauchy problem for lower semicontinuous differential inclusions

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A boundary value problem for implicit vector differential inclusions without assumptions of lower semicontinuity

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Electronic structure and exchange constants in magnetic semiconductors digital alloys: chemical and band-gap effects

First-principles simulations have been performed for [001]-ordered Mn/Ge and Mn/GaAs "digital alloys", focusing on the effects of i) a larger band-gap and ii) a different semiconducting host on the electronic structure of the magnetic semiconductors of interest. Our results for the exchange constants in Mn/Ge, evaluated using a frozen-magnon scheme, show that a larger band-gap tends to give a stronger nearest-neighbor ferromagnetic coupling and an overall enhanced in-plane ferromagnetic coupling even for longer-ranged coupling constants. As for the chemical effects on the exchange constants, we show that Mn/GaAs shows a smaller nearest-neighbor ferromagnetic coupling than Mn/Ge, but exchange constants for higher Mn-Mn distance show an overall increased ferromagnetic behavior in Mn/GaAs. As a result, from the magnetic-coupling point of view, the two systems behave on average rather similarly

Structure of the silicon vacancy in 6H-SiC after annealing identified as the carbon vacancy–carbon antisite pair

We investigated radiation-induced defects in neutron-irradiated and subsequently annealed 6H-silicon carbide (SiC) with electron paramagnetic resonance (EPR), the magnetic circular dichroism of the absorption (MCDA), and MCDA-detected EPR (MCDA-EPR). In samples annealed beyond the annealing temperature of the isolated silicon vacancy we observed photoinduced EPR spectra of spin S=1 centers that occur in orientations expected for nearest neighbor pair defects. EPR spectra of the defect on the three inequivalent lattice sites were resolved and attributed to optical transitions between photon energies of 999 and 1075 meV by MCDA-EPR. The resolved hyperfine structure indicates the presence of one single carbon nucleus and several silicon ligand nuclei. These experimental findings are interpreted with help of total energy and spin density data obtained from the standard local-spin density approximation of the density-functional theory, using relaxed defect geometries obtained from the self-consistent charge density-functional theory based tight binding scheme. We have checked several defect models of which only the photoexcited spin triplet state of the carbon antisite–carbon vacancy pair (CSi-VC) in the doubly positive charge state can explain all experimental findings. We propose that the (CSi-VC) defect is formed from the isolated silicon vacancy as an annealing product by the movement of a carbon neighbor into the vacancy

Parametrization of Riemann-measurable selections for multifunctions of two variables with application to differential inclusions

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Non-autonomous vector integral equations with discontinuous right-hand side

summary:We deal with the integral equation $u(t)=f(t,\int_I g(t,z)u(z)\,dz)$, with $t\in I:=[0,1]$, $f:I\times \Bbb R^n \to \Bbb R^n$ and $g:I\times I\to[0,+\infty[$. We prove an existence theorem for solutions $u\in L^s(I,\Bbb R^n)$, $s\in \,]1,+\infty]$, where $f$ is not assumed to be continuous in the second variable. Our result extends a result recently obtained for the special case where $f$ does not depend explicitly on the first variable $t\in I$