we consider the number of zeros of the integral $i(h) = oint_{gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. we prove that the number of zeros of $i(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.

1 Recap Recall some important facts about entropy and mutual information from the previous lecture: • H(X,Y ) = H(X) + H(Y |X) = H(Y ) + H(X|Y ) • I(X;Y ) = H(X)−H(X|Y ) = H(Y )−H(Y |X) = H(X) + H(Y )−H(X,Y ) • I(X;Y |Z) = H(X|Z)−H(X|Y,Z) • I(X;Y ) = 0 if X and Y are independent • I(X;Y ) ≥ 0 or, equivalently, H(X) ≥ H(X|Y ) Exercise 1.1 Prove that H(X|Y ) = 0 if and only if X = g(Y ) for some ...

BACKGROUND Approximately half recurrent miscarriage (RM) cases remain unexplained after standard investigations. Secondary RM (SRM) is, in contrast to primary RM, preceded by a birth, which increases the transfer of fetal cells into the maternal circulation. Mothers of boys are often immunized against male-specific minor histocompatibility (H-Y) antigens, and H-Y immunity can cause graft-versus...

In men, sexual orientation correlates with the number of older brothers, each additional older brother increasing the odds of homosexuality by approximately 33%. It is hypothesized that this fraternal birth order effect reflects the progressive immunization of some mothers to Y-linked minor histocompatibility antigens (H-Y antigen) by each succeeding male fetus, and the concomitantly increasing...

In order to test the microwave properties of superconductors, we are developing a disk resonator, to be operated in the TM010 mode at 3.36 GHz. The goal is a measurement of microwave surface resistance at both low and high power, initially for niobium thin films, but eventually for other thin film superconductors. The currents in a disk resonator flow in and out radially. Edge currents are zero...