Math 131: Introduction to Topology 1
11/6/2019 - The Brouwer Fixed Point Theorem85 11/11/2019 - Antipodes and the Borsuk-Ulam Theorem88 11/13/2019 - Deformation Retracts and Homotopy Equivalence91 11/18/2019 - Computing the Fundamental Group95 11/20/2019 - Equivalence of Covering Spaces and the Universal Cover99 11/25/2019 - Universal Covering Spaces, Free Groups104
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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. 13. Cryptology An integer p>1 is primeif 1 and pare the only factors of p. The number kmod pis the ...
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.
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 …
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 ...
he walked half of the remaining distance, so now he was 3/4 of the way to the grocery. In the following ten minutes he walked half of the remaining ... of the nautilus shell, the number of seeds in consecutive rows of a sunﬂower, and many natural …
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;
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!
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
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.
More generally, when the axis is not fixed, ... r = "lever arm" = distance from axis to point of ... Parallel Axis Theorem Relates I cm (axis through center-of-mass) to I w.r.t. some other axis: I = I cm + M d 2 (See proof in appendix.) Example: Rod of length L, mass M 2 CM 1
as the theorem itself. 1.2 The geometric version of the CBH formula. To state this formula we introduce some notation. Let ad Adenote the operation of bracketing on the left by A, so adA(B) := [A,B]. Deﬁne the function ψby ψ(z) = zlogz z−1 which is deﬁned as a convergent power series around the point z= 1 so ψ(1+u) = (1+u) log(1+u) u ...