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Consider the one-dimensional Schrödinger equation (1) i ∂u ∂t = ∂ 2 u ∂x 2 , with solution operator e −it∆. If u(t, x) is defined on R × S 1 , where S 1 = R/2πZ is the unit circle, then e −it∆ has a very complicated behavior. It turns out that, if t is a rational multiple of 2π, then (2) S(t, x) = e −it∆ δ(x) can be written as a finite linear combination of(More)
CONTENTS Introduction. 0. Pseudodifferential operators and linear PDE. §0.1 The Fourier integral representation and symbol classes §0.2 Schwartz kernels of pseudodifferential operators §0.3 Adjoints and products §0.4 Elliptic operators and parametrices §0.5 L 2 estimates §0.6 Gårding's inequality §0.7 The sharp Gårding inequality §0.8 Hyperbolic evolution(More)
Chronic unpredictable mild stress (CMS), an animal model of depression, downregulates hippocampal CB1 receptors in adult male rats. Given that endocannabinoids are implicated in modulating stress and anxiety and that women are vulnerable to stress-related disorders, we tested the effects of CMS on both female and male rats. Gonadectomized (gndx) and(More)
Steroid-mediated sexual differentiation of the brain is a developmental process that permanently organizes the brain into a male or female phenotype. Previous studies in the rodent have examined the steroid-mediated mechanisms of male brain development. In an effort to identify molecules involved in female brain development, a high-throughput proteomics(More)
Discrete numerical approximations with additional conserved quantities are developed here both for barotropic geophysical flows generalizing the 2D incompressible fluid equations and suitable discretizations of the Burgers-Hopf equation. Mathematical, numerical, and statistical properties of these approximations are studied below in various different(More)
To predict the location of natural resources and reduce the cost of exploration, geophysicists rely on various techniques to map the internal structure of the earth. One common mapping method probes the earth's interior using an acoustic energy source (sound waves). The acoustic waves reeect when they impinge on a location where the acoustic velocity eld(More)
1 2 Contents 0. One-variable calculus 1. The derivative 2. Inverse function and implicit function theorem 3. Fundamental local existence theorem for ODE 4. The Riemann integral in n variables 5. Integration on surfaces 6. Differential forms 7. Products and exterior derivatives of forms 8. The general Stokes formula 9. The classical Gauss, Green, and Stokes(More)