Camas School District; Mathematics Curriculum, Student Learning Objectives K-8

The Camas School District; Student Learning Objectives (K-8) is designed to help school district personnel comply with Washington State Student Learning Objectives Law RCW 28A.58.090, which requires that all school districts in the state of Washington develop student learning objectives in the areas of reading, language arts, and mathematics. This guide was developed to identify, clarify, and catagorize, sets of goals and student learning objectives which will improve students\u27 learning opportunities in the area of mathematics, and at the same time provide direction for the classroom teacher. This guide does not reflect all of the goals and objectives taught at any specific grade level, but the goals and objectives which the community of Camas and the Camas School District feel are the backbone of our mathematics curriculum. This guide will be considered a working document. As it is used, suggestions for change or revision will be noted so that the guide can constantly reflect the views of the Camas community, school district and the needs of our students. An effort was made by the community and professional staff not to develop the goals and objectives around any one set of materials. The scope and sequence should prove to be appropriate for a variety of materials. The purpose of this document is to satisfy the legal requirements of the Student Learning Objectives Law RCW 28A.58.090 and to provide the Camas School District with math student learning objectives and indicators. Contained in this guide are the objectives and indicators that the Camas School District chose as their student learning objectives. Through the use of these objectives the parents, the Camas School District, and the Superintendent of Public Instruction will be able to measure student achievement and performance

An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary

We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed

The semiclassical theory of discontinuous systems and ray-splitting billiards

We analyze the semiclassical limit of spectral theory on manifolds whose metrics have jump-like discontinuities. Such systems are quite different from manifolds with smooth Riemannian metrics because the semiclassical limit does not relate to a classical flow but rather to branching (raysplitting) billiard dynamics. In order to describe this system we introduce a dynamical system on the space of functions on phase space. To identify the quantum dynamics in the semiclassical limit we compute the principal symbols of the Fourier integral operators associated to reflected and refracted geodesic rays and identify the relation between classical and quantum dynamics. In particular we prove a quantum ergodicity theorem for discontinuous systems. In order to do this we introduce a new notion of ergodicity for the ray-splitting dynamics

Quantitative Determination of Temperature in the Approach to Magnetic Order of Ultracold Fermions in an Optical Lattice

We perform a quantitative simulation of the repulsive Fermi-Hubbard model using an ultracold gas trapped in an optical lattice. The entropy of the system is determined by comparing accurate measurements of the equilibrium double occupancy with theoretical calculations over a wide range of parameters. We demonstrate the applicability of both high-temperature series and dynamical mean-field theory to obtain quantitative agreement with the experimental data. The reliability of the entropy determination is confirmed by a comprehensive analysis of all systematic errors. In the center of the Mott insulating cloud we obtain an entropy per atom as low as 0.77k(B) which is about twice as large as the entropy at the Neel transition. The corresponding temperature depends on the atom number and for small fillings reaches values on the order of the tunneling energy

Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems

We show in this article that the Reeh-Schlieder property holds for states of quantum fields on real analytic spacetimes if they satisfy an analytic microlocal spectrum condition. This result holds in the setting of general quantum field theory, i.e. without assuming the quantum field to obey a specific equation of motion. Moreover, quasifree states of the Klein-Gordon field are further investigated in this work and the (analytic) microlocal spectrum condition is shown to be equivalent to simpler conditions. We also prove that any quasifree ground- or KMS-state of the Klein-Gordon field on a stationary real analytic spacetime fulfills the analytic microlocal spectrum condition.Comment: 31 pages, latex2

Estimating the nuclear level density with the Monte Carlo shell model

A method for making realistic estimates of the density of levels in even-even nuclei is presented making use of the Monte Carlo shell model (MCSM). The procedure follows three basic steps: (1) computation of the thermal energy with the MCSM, (2) evaluation of the partition function by integrating the thermal energy, and (3) evaluating the level density by performing the inverse Laplace transform of the partition function using Maximum Entropy reconstruction techniques. It is found that results obtained with schematic interactions, which do not have a sign problem in the MCSM, compare well with realistic shell-model interactions provided an important isospin dependence is accounted for.Comment: 14 pages, 3 postscript figures. Latex with RevTex. Submitted as a rapid communication to Phys. Rev.

Maternal One-Carbon Nutrient Intake and Risk of Being Overweight or Obese in Their Offspring: A Transgenerational Prospective Cohort Study

Adherence to healthful dietary patterns is associated with lower body mass index (BMI) in adults; however, whether maternal diet quality during peripregnancy is related to a lower overweight risk in the offspring remains to be elucidated. We investigated the associations between the Alternate Healthy Eating Index (AHEI), Alternate Mediterranean Diet (aMED) and Dietary Approach to Stop Hypertension (DASH) during peripregnancy and offspring weight outcomes in a study including 2729 mother-child pairs from the Nurses' Health Study II and offspring cohort Growing Up Today Study II. Children, 12-14 years at baseline were 21-23 years at the last follow-up. Overweight or obesity was defined according to International Obesity Task Force (< 18 years) and World-Health-Organization guidelines (18 + years). Maternal dietary patterns were calculated from food frequency questionnaires. Log-binomial models were used to estimate relative risks (RR) and 95% confidence intervals. In models adjusted for sex, gestational age at delivery and maternal total energy intake, greater maternal adherence to aMED and DASH, but not AHEI, was associated with lower overweight risk in the offspring (RRQ5 vs Q1 = 0.82 [0.70-0.97] for aMED and 0.86 [0.72-1.04] for DASH, P for trend < 0.05 for both). After additional adjustment for maternal pre-pregnancy lifestyle factors and socio-demographic characteristic, none of the diet quality scores were significantly associated with offspring overweight risk. Maternal pre-pregnancy BMI did not modify any of these associations. In this population of generally well-nourished women, maternal healthful dietary patterns during the period surrounding pregnancy were not independently associated with offspring overweight risk at ages 12-23 years

Local covariant quantum field theory over spectral geometries

A framework which combines ideas from Connes' noncommutative geometry, or spectral geometry, with recent ideas on generally covariant quantum field theory, is proposed in the present work. A certain type of spectral geometries modelling (possibly noncommutative) globally hyperbolic spacetimes is introduced in terms of so-called globally hyperbolic spectral triples. The concept is further generalized to a category of globally hyperbolic spectral geometries whose morphisms describe the generalization of isometric embeddings. Then a local generally covariant quantum field theory is introduced as a covariant functor between such a category of globally hyperbolic spectral geometries and the category of involutive algebras (or *-algebras). Thus, a local covariant quantum field theory over spectral geometries assigns quantum fields not just to a single noncommutative geometry (or noncommutative spacetime), but simultaneously to ``all'' spectral geometries, while respecting the covariance principle demanding that quantum field theories over isomorphic spectral geometries should also be isomorphic. It is suggested that in a quantum theory of gravity a particular class of globally hyperbolic spectral geometries is selected through a dynamical coupling of geometry and matter compatible with the covariance principle.Comment: 21 pages, 2 figure

Quantum Energy Inequalities in Pre-Metric Electrodynamics

Pre-metric electrodynamics is a covariant framework for electromagnetism with a general constitutive law. Its lightcone structure can be more complicated than that of Maxwell theory as is shown by the phenomenon of birefringence. We study the energy density of quantized pre-metric electrodynamics theories with linear constitutive laws admitting a single hyperbolicity double-cone and show that averages of the energy density along the worldlines of suitable observers obey a Quantum Energy Inequality (QEI) in states that satisfy a microlocal spectrum condition. The worldlines must meet two conditions: (a) the classical weak energy condition must hold along them, and (b) their velocity vectors have positive contractions with all positive frequency null covectors (we call such trajectories `subluminal'). After stating our general results, we explicitly quantize the electromagnetic potential in a translationally invariant uniaxial birefringent crystal. Since the propagation of light in such a crystal is governed by two nested lightcones, the theory shows features absent in ordinary (quantized) Maxwell electrodynamics. We then compute a QEI bound for worldlines of inertial `subluminal' observers, which generalizes known results from the Maxwell theory. Finally, it is shown that the QEIs fail along trajectories that have velocity vectors which are timelike with respect to only one of the lightcones