Theory of functions of a real variable.

In Chapter II I do the basics of Hilbert space theory, i.e. what I can do without measure theory or the Lebesgue integral. The hero here (and perhaps for the first half of the course) is the Riesz representation theorem. Included is the spectral theorem for …

  Basics

A Mathematical Theory of Communication

J. W. Tukey. A device with two stable positions, such as a relay or a flip-flop circuit, can store one bit of information. N such devices can store N bits, since the total numberof possible states is 2N and log 2 2 N = N. If the base 10 is used the units may be called decimal digits. Since log2 M = log10 M log10 2 = 3: 32log10 M;

  Devices, Early

SOME FUNDAMENTAL THEOREMS IN MATHEMATICS

FUNDAMENTAL THEOREMS Theorem: I(V(J)) = p J. The theorem is due to Hilbert. A simple example is when J= hpi= hx2 2xy+ y2iis the idealJgeneratedbypinR[x;y];thenV(J) = fx= ygandI(V(J)) istheidealgeneratedby x y. Forliterature,see[294]. 13. Cryptology An integer p>1 is primeif 1 and pare the only factors of p. The number kmod pis the ...

  Fundamentals, Theorem

Group Theory and the Rubik's Cube

Again, we’re going to rewrite this using new symbols. Let mean multiplication, and let e= 1, a= 2, b= 4, and c= 3. Then, the multiplication table for (Z=5Z) looks like e a b c e e a b c a a b c e b b c e a c c e a b Notice that this is exactly the same as the table for addition on Z=4Z!

  Table, Multiplication, Multiplication table

Lie algebras - people.math.harvard.edu

8 CHAPTER 1. THE CAMPBELL BAKER HAUSDORFF FORMULA A+B+ 1 2 A2 +AB+ 1 2 B2 − 1 2 (A+B+···)2 = A+B+ 1 2 [A,B]+··· where [A,B] := AB−BA (1.1) is the commutator of Aand B, also known as the Lie bracket of Aand B.

  Algebra

Unit 5: Change of Coordinates

B= S 1v. Theorem: If T(x) = Ax is a linear map and S is the matrix from a basis change, then B = S 1AS is the matrix of T in the new basis B. Proof. Let y = Ax. The statement [y] B= B[x] Bcan be written using the last theorem as S 1y = BS 1x so that y = SBS 1x. Combining with y = Ax, this gives B = S 1AS. 5.4. If two matrices A;B satisfy B = S ...

Topology - people.math.harvard.edu

theory. Part II is an introduction to algebraic topology, which associates algebraic structures such as groups to topological spaces. We will follow Munkres for the whole course, with some occassional added topics or di erent perspectives. We will consider topological spaces axiomatically. That is, a topological

  Theory, Algebraic

Higher Algebra - people.math.harvard.edu

to Yoneda’s lemma, this property determines the space Zup to homotopy equivalence. Moreover, since the functor X7!K(X) takes values in the category of commutative rings, the topological space Z is automatically a commutative ring object in the homotopy category H of topological spaces.

  Yoneda

Lines and Planes in R3

Lines and Planes in R3 A line in R3 is determined by a point (a;b;c) on the line and a direction ~v that is parallel(1) to the line. The set of points on this line is given by fhx;y;zi= ha;b;ci+ t~v;t 2Rg This represents that we start at the point (a;b;c) and add all scalar multiples of the vector ~v.

  Line, Points, Point of

Lecture 9: Partial derivatives

Lecture 9: Partial derivatives If f(x,y) is a function of two variables, then ∂ ∂x f(x,y) is defined as the derivative of the function g(x) = f(x,y), where y is considered a constant. It is called partial derivative of f with respect to x. The partial derivative with respect to y is defined similarly. We also use the short hand notation ...

  Derivatives

Foot and Ankle Conditioning Program

Equipment needed: 20 marbles Step-by-step directions • Sit with both feet flat and place 20 marbles on the floor in front of you. • Use your toes to pick up one marble at a time and place into a bowl. • Repeat until you have picked up all the marbles. Foot and Ankle Conditioning Program STRENGTHENING EXERCISES Repetitions 20 Days per week ...

  Exercise, Ankle, Ambler

Chapter 13 NUCLEAR FISSION

of marbles on a roller coaster (Figure 13-2). The path of the fission reaction is from left to right. From bottom to top, the energy of the reacting atoms (or marbles) increases; also, when these particles are in the valleys, they are relatively stable, and …

  Chapter, Nuclear, Ambler, Fission, Chapter 13 nuclear fission

Conditional Probability

blue marbles, I draw a marble at random from the bag and then, without replacing the rst marble, I draw a second marble. What is the probability that both marbles are red? We can draw a tree diagram to represent the possible outcomes of the above experiment and label it with the

  Probability, Ambler, Conditional, Conditional probability

Solar System Scale Model

Extra Earth marbles (optional) Eclipse glasses (optional) Printed satellite image of event location (optional) Transparent overlay of scaled planet orbits (optional) Background: The scale of our solar system is difficult to imagine when we are standing on what appears to be a large

  System, Model, Scale, Solar, Ambler, Solar system scale model

Fire Safety - U.S. Scouting Service Project

Children are taught not to give marbles, etc. to younger children. Toys are sturdy, do not come apart easily, & have no sharp edges or points. Nontoxic paint is used on baby furniture and toys. All houseplants are nonpoisonous.

  Safety, Fire, Ambler, Fire safety

Porosity and Bulk Density of Sedimentary Rocks

marbles of the United States and derived total porosity by the grain density-bulk density relation. More recently Melcher (1924) gave a critical discussion of previous methods of porosity determination and presented many determina­ tions of bulk density and total porosity of oil sands obtained by the

  Ambler, Porosity

Upper Body: Fine Motor Exercise Program Finger Exercises ...

Flick small objects with each finger (marbles, paperclips, cotton balls, etc.) Practice taking tacks or pins off a bulletin board and putting them back on or put pins into a Styrofoam ball or cup. Open and close safety pins of different sizes. Gather small items such as pins, buttons, coins, toothpicks, paper clips, rice, beans,

  Ambler

Distributive Property Multiplication

5 Below is a list of common misconceptions and questions/prompts that may be written on individual tasks, on the board or asked during the collaborative activity to help students clarify and extend their thinking.

  Property, Multiplication, Distributive, Distributive property multiplication