Consider the ising ferromagnet, given by the spin hamiltonian h--jσσσ; - η σσ,. (ij) i-1 where j> 0, ai= t1 , and the first sum is over nearest neighbors on a simple triangular lattice. use the real-space renormalization-group scheme of niemeyer and van leeuwen, up to first-order terms, as dis- cussed in the last section, to obtain the thermal and magnetic critical exponents. consider the ising ferromagnet, given by the hamiltonian n m= -jσσ.σ; -η σσ, i=1 (ij) where j> 0, oi = ±1, and the first sum is over nearest-neighbor sites on a square lattice. choose a unit cell consisting of five spins, as indicated in figure 14.9. use the real-space renormalization-group scheme of niemeyer and van leeuwen to write a set of recursion relations up to first-order terms in the interactions. obtain the values of the thermal and magnetic critical exponents.

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Follow these directions and answer the questions. 1. shine a pencil-thin beam of light on a mirror perpendicular to its surface. (if you don't have a laser light as suggested in the video, you can make a narrow beam from a flashlight by making a cone from black construction paper and taping it over the face of the flashlight.) how does the light reflect? how does the relationship of incident to reflected ray relate to the reflection of water waves moving perpendicular to a barrier? 2. shine a pencil-thin beam of light on a mirror standing on a sheet of paper on the table (or floor) so that you can mark the incident ray and reflected ray. (you can support the mirror from the back by taping it to a wooden block.) 3. mark a line on the paper representing the reflective surface. (the reflective surface of a mirror is usually the back edge.) 4. draw a dashed line perpendicular to the mirror surface at a point where the incident and reflected ray meet. this perpendicular is called a normal to the surface. 5. measure the angles between the rays and the normal. the angle of incidence is the angle formed by the incident ray and the normal to the surface. the angle formed by the reflected ray and normal is called the angle of reflection (r). what is the angle of incidence? what is the angle of reflection? 6. repeat for several different angles. (see report sheet for details.) what appears to be the relationship between the angle of incidence and angle of reflection? in science 1204, what was the relationship for these two angles made by the reflection of waves in a ripple tank? 7. roll a ball bearing so that it hits a fixed, hard surface (a metal plate) at several angles (including head-on). observe the way in which the ball bearing reflects. what generalization can you make about how a ball bearing reflects from a wall? have you proved that light can only behave like a wave?

Assume similar data for the motion of the blood in your aorta. estimate how many beats of the heart it will take the blood to get from your heart to your brain. (assume that the distance from your heart to your brain is 30 cm)