Vanderbilt - CHEM - 220A

Vanderbilt - CHEM - 220A

Solving Combined Spectra Problems: Mass Spectra: Molecular Formula Nitrogen Rule # of nitrogen atoms in the molecule M+1 peak # of carbons Degrees of Unsaturation: # of rings and/or -bonds Infrared Spectra Functional Groups C=O C=C CC 1H NMR: Chemical Shi

Vanderbilt - CHEM - 220A

Writing Chemical StructuresH H C H H H H H C C C C H H H H Kekule Condensed H CH3 CH3CH2CHCH3 CH3CH2CH(CH3)2SkeletalSkeletal Notation 1. 2. 3. 4. 5. Carbon atoms are at the ends of lines and at the intersection of two lines Hydrogens on carbon atoms ar

Vanderbilt - CHEM - 220A

STEREOCHEMICAL REPRESENTATION Stereochemistry is an important aspect of Organic Chemistry and you must be able to draw structures which clearly indicate the stereochemical orientation of the various groups on a molecule. For any given compound there will

Vanderbilt - CHEM - 220A

Some Organic Synthesis Practice Problems: Starting from 1-hexene, 1-butyne, bromoethane, iodomethane and any reagent needed (you do not need to use all of these compounds), synthesize: 1.O2.O H3. octane4.BrBr (meso)5Cl Cl6OHOH (racemic)

Vanderbilt - CHEM - 220A

Some Organic Synthesis Practice Problems: Starting from 1-hexene, 1-butyne, bromoethane, iodomethane and any reagent needed (you do not need to use all of these compounds), synthesize: 1.O H3CCO3H H2, Lindlar's Catalyst H3C C C CH2CH3 NaNH2, NH3, CH3I H

Vanderbilt - CHEM - 220A

Cahn-Ingold-Prelog Priority Rules 1. Look at the four atoms directly attached to the stereogenic center (X). Assign priorities based on atomic number to all four atoms. Priority 1 is assigned to the atom or group of highest atomic number, priority 4 to th

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 4: Elastic PropertiesElastic constants Elastic properties of solids are determined by interatomic forces acting on atoms when they are displaced from the equilibrium positions. At small deformations these forces are propor

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 1: Crystal StructureA solid is said to be a crystal if atoms are arranged in such a way that their positions are exactly periodic. This concept is illustrated in Fig.1 using a two-dimensional (2D) structure. yTC Fig.1A

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 2: X-ray Diffraction and Reciprocal LatticeBragg law. Most methods for determining the atomic structure of crystals are based of the idea of scattering of radiation. X-rays is one of the types of the radiation which can be

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 3: Crystal BindingInteratomic forces Solids are stable structures, and therefore there exist interactions holding atoms in a crystal together. For example a crystal of sodium chloride is more stable than a collection of fr

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 4: Elastic PropertiesElastic constants Elastic properties of solids are determined by interatomic forces acting on atoms when they are displaced from the equilibrium positions. At small deformations these forces are propor

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 5: Lattice VibrationsSo far we have been discussing equilibrium properties of crystal lattices. When the lattice is at equilibrium each atom is positioned exactly at its lattice site. Now suppose that an atom displaced fro

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 6: Thermal propertiesHeat Capacity There are two contributions to thermal properties of solids: one comes from phonons (or lattice vibrations) and another from electrons. This section is devoted to the thermal properties o

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 7: Free electron modelA free electron model is the simplest way to represent the electronic structure of metals. Although the free electron model is a great oversimplification of the reality, surprisingly in many cases it

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 8: Electronic TransportDrude model The simplest treatment of the electrical conductivity was given by Drude. There are four major assumptions within the Drude model. 1. Electrons are treated as classical particles within a

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 9: Energy bandsThe free electron model gives us a good insight into many properties of metals, such as the heat capacity, thermal conductivity and electrical conductivity. However, this model fails to help us other importa

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 10 Metals: Electron Dynamics and Fermi SurfacesElectron dynamics The next important subject we address is electron dynamics in metals. Our consideration will be based on a semiclassical model. The term "semiclassical" come

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 11: Methods for calculating band structureThe computational solid state physics is a very fast growing area of research. Modern methods for calculating the electronic band structure of solids allow predicting many importan

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 12: SemiconductorsCrystal structure and bonding Semiconductors include a large number of substances of widely different chemical and physical properties. These materials are grouped into several classes of similar behavior

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 13: Optical properties of solidsOptical methods are very useful for the quantitative determination of the electronic band structure of solids. Experiments on optical reflectivity, transmission and refraction provide the wa

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 14: Dielectric properties of insulatorsThe central quantity in the physics of dielectrics is the polarization of the material P. The polarization P is defined as the dipole moment p per unit volume. The dipole moment of a

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 15: Magnetic properties of materialsDefinition of fundamental quantities When a material medium is placed in a magnetic field, the medium is magnetized. This magnetization is described by the magnetization vector M, the di

University of Nebraska - PHYSICS - 927

Physics 927 E.Y.TsymbalSection 16: Magnetic properties of materials (continued)Ferromagnetism Ferromagnetism is the phenomenon of spontaneous magnetization the magnetization exists in the ferromagnetic material in the absence of applied magnetic field.

ASU - MAT - 473

MAT 47311. Euclidean Space, Linear Maps. f (x + h) - f (x) h0 h exists. When it does, we call the value of the limit the derivative of f at x, and denote it by f (x). Equivalently, f is differentiable at x if there exists c R such that lim lim Recall th

ASU - MAT - 473

4SPRING 20122. Linear Maps, Operator Norm It's not obvious from the definition that T is finite for all T L(Rn , Rm ). If [T ] = (aij ) is the matrix representation of T with respect to the standard bases, define the 2-norm of T to be m n 2 T 2 = aij .

ASU - MAT - 473

36SPRING 201214. Measurability Criteria Theorem 14.1. L is a -algebra of subsets of Rn , and is a measure on L. Proof. L0 , so L. If A L, then Ac M = M \A = M \(AM ) L0 for each M L0 , so Ac L. Finally, if Ai L for each i N and A = i Ai , then (A M ) (M

ASU - MAT - 473

MAT 47394. Chain Rule. Partial Derivatives. Proposition 4.1 (Chain Rule). Suppose f : U Rn Rm is differentiable at x U and g : V Rm R is differentiable at f (x) V . Then g f is differentiable at x, with (g f ) (x) = g (f (x) f (x). Proof. Put y = f (x)

ASU - MAT - 473

MAT 4733916. Measurable Functions Recall that the -algebra B of Borel sets in R is the smallest -algebra of subsets of R which contains the open sets. Definition 16.1. Let M be a -algebra of subsets of a set X. A function f : X R is said to be M-measura

ASU - MAT - 473

MAT 4732711. Lebesgue Measure: Closed Boxes and Special Polygons Recall that relatively few functions f : R R are Riemann integrable: the continuous functions, and in a sense not too many more. Also, the Riemann integral behaves badly with respect to li

ASU - MAT - 473

MAT 47339 15. Caratheodory's Theorem; Nonmeasurable and Non-Borel Sets Theorem 15.1 (Carathodory). A RN is measurable if and only if e for each E RN . (E) = (E A) + (E Ac )Lemma 15.2. For every measurable set B and every A B, Exercise 15.1. Prove Lemma

ASU - MAT - 473

42SPRING 201217. Measurable Functions, Simple Functions Proposition 17.1. If f, g : X R are M-measurable, then f +g and f g are M-measurable. Proof. For each t R, cfw_x | f (x) + g(x) < t = cfw_x | f (x) < r cfw_x | g(x) < t - r .rQSince f and g are M

ASU - MAT - 473

MAT 473177. Taylor's Theorem; The Inverse Function Theorem. Theorem 7.1 (Taylor's Theorem). Let U Rn be open and convex, and let f : U R be C N . For each a U and each h such that a + h U , there exists c on the line segment from a to a + h such thatf

ASU - MAT - 473

20SPRING 20128. Proof of The Inverse Function Theorem Proof. By Exercises 7.3 and 7.4, we may assume that a = 0, f (0) = 0, and f (0) = Id. Now, since f is continuous on U , (f - Id) = f - Id is, so there exists > 0 such that (f - Id) (x) - (f - Id) (0)

ASU - MAT - 473

22SPRING 2012 Consider f : R2 R defined by f (x, y) = x2 + y 2 - 1, which is C 1 with f (x, y) = 2x 2y for all (x, y). The level set cfw_(x, y) | f (x, y) = 0 is a nice curve in R2 , namely the unit circle x2 + y 2 = 1. Moreover, if we look near the poi

ASU - MAT - 473

30SPRING 201212. Lebesgue Measure: Open Sets and Compact Sets So far we have defined (A) for A empty, for any closed box A, and more generally for any special polygon A Rn . Definition 12.1. If G Rn is a nonempty open set, let (G) = supcfw_(P ) | P is a

ASU - MAT - 473

6SPRING 20123. Invertible Operators, Derivatives For each T L(Rn ), let det T be the determinant of the n n matrix representing T with respect to the standard basis; this defines a map det : L(Rn ) R. In fact, det is continuous. To see this, note that t

ASU - MAT - 473

24SPRING 201210. Proof of the Implicit Function Theorem. Proof. The formula for g (x) will follow from the rest of the theorem, as in Exercise 9.1, by defining : Rn Rn Rm by (x) = (x, g(x), so that (f )(x) = f (x, g(x) = 0 for all x I, and then using th

ASU - MAT - 473

MAT 4733313. Lebesgue measure: inner and outer measures Proposition 13.1. Let Ai be subsets of RN . (i) If A2 , then (A1 ) (A2 ) and (A1 ) (A2 ). A1 (ii) i=1 Ai i=1 (Ai ). (iii) If the Ai are disjoint, then Ai (Ai ). i=1 i=1Proof. (i) is routine. For (

ASU - MAT - 473

12SPRING 20125. Partial Derivatives, continued The extra condition on f that provides a converse for Proposition 4.5 is the continuity of its partial derivatives, in the sense that x Dj fi (x) should be continuous from U Rn into R for each i, j. (See Pr

ASU - MAT - 473

14SPRING 20126. Mean Value Theorem. Clairaut's Theorem Recall the Mean Value Theorem from Calculus: if f : [a, b] R is continuous on [a, b] and differentiable on (a, b), then there exists c between a and b such that Disappointingly, we can have no such

ASU - MAT - 473

ASSIGNMENT 2 SOLUTIONSMAT 473 SPRING 2012Exercise 3.2.then T GL(n). (ii) Explain how it follows from this that GL(n) is an open subset of L(Rn ). Proof. For (i), apply Proposition 3.2 to T S -1 : T S -1 - Id = (T - S) S -1 T - SS -1 < 1, cfw_T L(Rn ) |

ASU - MAT - 473

ASSIGNMENT 7 SOLUTIONSMAT 473 SPRING 2012Exercise 12.2. Prove that if x Rn and P is a special polygon in Rn , then (P ) = (P +x). Conclude that (G) = (G + x) for every open set G in Rn . Solution. It is clear that I +x is a closed box if and only if I i

ASU - MAT - 473

ASSIGNMENT 6 SOLUTIONSMAT 473 SPRING 2012Exercise 10.3. The equation x3 + y 3 + z 3 - 3xyz = 0 can be solved for z as a function of (x, y) near (-1, -1, 2). If z = g(x, y) denotes such a function, find the degree 2 Taylor polynomial of g for (x, y) near

ASU - MAT - 473

ASSIGNMENT 5 SOLUTIONSMAT 473 SPRING 2012Exercise 8.1. Suppose U Rn is open and f, g : U Rm are continuously differentiable on U . Prove that the function : U R defined by (x) = f (x), g(x) is continuously differentiable on U . Proof. By the Product Rul

ASU - MAT - 473

ASSIGNMENT 4 SOLUTIONSMAT 473 SPRING 2012Exercise 4.2. Consider the function f : R2 R defined by 2 2 xy x - y if (x, y) = (0, 0) f (x, y) = x2 + y 2 0 if (x, y) = (0, 0).Prove that all second-order partial derivatives of f exist, but the conclusion to

ASU - MAT - 473

ASSIGNMENT 9 SOLUTIONSMAT 473 SPRING 2012Exercise 16.2. Prove that each of the following collections generates the -algebra B of Borel sets in R: (ii) cfw_(a, b] | a < b in R, and (vii) cfw_[t, +) | t R. Proof. Let C be the collection in part (ii), and

ASU - MAT - 473

ASSIGNMENT 8 SOLUTIONSMAT 473 SPRING 2012Exercise 14.1. For A Rn and x Rn , prove that A + x L if and only if A L, in which case (A + x) = (A). Proof. For any x Rn , translation by x is continuous on Rn with continuous inverse (namely, translation by -x

ASU - MAT - 473

ASSIGNMENT 1 SOLUTIONSMAT 473 SPRING 2012Problem 1 (1.1). Given that R is complete, prove that Rn is complete (as a metric space). Proof. First note that for any y = (y1 , y2 , . . . , yn ) Rn , and for any 1 i n, we have 2 2 2 |yi | y1 + y2 + + yn = y

ASU - MAT - 473

ASSIGNMENT 3 SOLUTIONSMAT 473 SPRING 2012Exercise 4.2. Suppose f : R2 R is defined by 2 xy if (x, y) = (0, 0) f (x, y) = x2 + y 2 0 if (x, y) = (0, 0).Show that all partial derivatives of f exist everywhere, and f is continuous everywhere, but f (0, 0)

ASU - MAT - 421

Numerical Simulation of High Mach Number Astrophysical Jets with Radiative Cooling Carl Gardner Arizona State University Youngsoo Ha (KAIST), Steve Dwyer, Chi-Wang Shu (Brown), Jeff Hester & Kevin Healy (ASU), John Krist & Karl Staplefeldt (JPL)3210-

ASU - MAT - 421

Numerical Methods for Boundary Value ProblemsBVPs are usually formulated for y(x). Along the x axis, allocate gridpoints xi , i = 0, . . . , N . BCs will be imposed at x0 and xN . First and Second Derivative Matrices First and second derivatives at the i

ASU - MAT - 421

EigenvaluesThe eigenvalue problem is Ax = x Eigenvalues can be calculated from detcfw_A - I = 0. Note that detcfw_A = 1 2 n , Trcfw_A = 1 + 2 + + nPower Method for EigenvaluesStart with any x and expand x = c1 x1 + c2 x2 + + cn xn in terms of the eigen

ASU - MAT - 421

Equivalence Theorem (Lax-Richtmyer)The Fundamental Theorem of Numerical Analysis. For consistent numerical approximations, stability and convergence are equivalent. Lax proved for IVPs. Applies as well to BVPs, approximations to functions and integrals,

ASU - MAT - 421

IEEE Floating PointA real number r is represented on the computer by r r = (1 + f ) 2n where the 1 is a phantom. In double precision floating point on a 32-bit word computer, there is one sign bit s (s = 0 for + and s = 1 for -), the mantissa 0 f < 1 is

ASU - MAT - 421

MAT 421 Applied Computational Methods Prof. Gardner (carl.gardner@asu.edu), Goldwater 654 Reading: Sections 1.7 (floating point) & 4.14.4 (finding roots) of Moler's Numerical Computing with MATLAB.Homework 1Due: Fri Jan 13 (1) Verify that the three-poin

ASU - MAT - 421

MAT 421 Applied Computational Methods Prof. Gardner (carl.gardner@asu.edu), Goldwater 654 Reading: Sections 1.1, 1.7, 4.14.4, 3.13.3 of Moler's Numerical Computing with MATLAB.Homework 2Due: Fri Jan 20 (1) Find a positive root of g(x) = exp(-x2 ) - 1 =0

ASU - MAT - 421

MAT 421 Applied Computational Methods Prof. Gardner (carl.gardner@asu.edu), Goldwater 654 Reading: Sections 1.1, 1.7, 4.14.4, 3.13.3, 6.16.4 of Moler's Numerical Computing with MATLAB.Homework 3Due: Fri Jan 27 (1) Problem 3.3 in Moler. Use interpolate.m

ASU - MAT - 421

MAT 421 Applied Computational Methods Prof. Gardner (carl.gardner@asu.edu), Goldwater 654 Reading: Sections 1.1, 1.7, 4.14.4, 3.13.3, 6.16.4, 7.17.4 of Moler's Numerical Computing with MATLAB.Homework 4Due: Fri Feb 3 (1) Derive Simpson's rule from the t

ASU - MAT - 421

MAT 421 Applied Computational Methods Prof. Gardner (carl.gardner@asu.edu), Goldwater 654 Reading: Sections 1.1, 1.7, 4.14.4, 3.13.3, 6.16.4, 7.17.4 of Moler's Numerical Computing with MATLAB.Homework 5Due: Fri Feb 10 (1) Verify the formulas in ivp1.m f

ASU - MAT - 421

MAT 421 Applied Computational Methods Prof. Gardner (carl.gardner@asu.edu), Goldwater 654 Reading: Sections 1.1, 1.7, 4.14.4, 3.13.3, 6.16.4, 7.17.4 of Moler's Numerical Computing with MATLAB.Homework 6Due: Fri Feb 17 (1) Prove that backward Euler is A-