An easily verifiable proof of the Brouwer fixed point theorem

We give a remarkably elementary proof of the Brouwer fixed point theorem. The proof is verifiable for most of the mathematicians.Comment: 4 page

Quantization of Conductance Minimum and Index Theorem

We discuss the minimum value of the zero-bias differential conductance $G_{\textrm{min}}$ in a junction consisting of a normal metal and a nodal superconductor preserving time-reversal symmetry. Using the quasiclassical Green function method, we show that $G_{\textrm{min}}$ is quantized at $(4e^2/h) N_{\mathrm{ZES}}$ in the limit of strong impurity scatterings in the normal metal. The integer $N_{\mathrm{ZES}}$ represents the number of perfect transmission channels through the junction. An analysis of the chiral symmetry of the Hamiltonian indicates that $N_{\mathrm{ZES}}$ corresponds to the Atiyah-Singer index in mathematics.Comment: 5 pages, 1 figur

An Elementary Proof of The 2-Dimensional Version of The Brouwer Fixed Point Theorem

We give an elementary proof of the 2-dimensional version of the Brouwer fixed point theorem. The proof can be extended to the n-dimensional version naturally. In our proof, we only use the Bolzano-Weierstrass theorem and the fact that an odd number plus an even number is odd