Shake and Bake Cancer

Cancer often spreads because the body's immune defenses somehow sleep through the invasion. A new technique that uses ultrasound may wake up the immune system to cancer wherever it may hide.

Duke University engineers and cancer doctors reported that certain frequencies of high intensity focused ultrasound shake cancer cells until they leak proteins. In lab animals those leaked proteins triggered strong immune system responses. The HIFU technique already heats cancer tumors to kill them. But shaking breaks cancer cell walls, and triggers an immune system attack on cancer cells, even those hiding in the blood stream or elsewhere in the body. Those so- called metastatic tumor cells are how cancer spreads even after surgery, chemotherapy and radiation. Doctors want to combine both the heating and shaking to up the odds of cancer survival.

Source: http://www.esa.int/esaCP/SEM9A3XAIPE_index_0.html

Geometric Construction of Roots of Quadratic Equation

A quadratic equation

with the leading coefficient a ≠ 0, has two roots that may be real - equal or different - or complex. The roots can be found from the quadratic formula:

In addition to the four arithmetic operations, the formula includes a square root. This is exactly the kind of operations that produce constructible numbers from constructible numbers. (By being "constructible" we mean a length (of a straight line) segment that could be constructed by straightedge and compass, given a unit length.) In fact, Descartes used this example to introduce his ideas of analytic geometry.

I came across a a generalization of Descartes' construction in an old translation (1910) of the classic The Theory of Geometric Constructions by August Adler (1863-1923).

(Read more...)

Source: http://www.cut-the-knot.org/changes.shtml