Sandy Aftermath Pictures from Upper Manhattan

Here are some pictures taken today in Manhattan during a quick bike ride, one day after Hurricane Sandy tore through New York City and the Northeast region.

Southbound on Lexington Ave, around E. 79th: hardly looks different

Northbound on Central Park West, around W. 72nd: destruction aftermath in view

Central Park West, around W. 82nd: downed tree

Hudson Greenway: open for a stretch

Hudson Greenway: closed at around 79th

Cherry Walk was anything but cherry-like, rather all muddy

Southbound Henry Hudson Parkway closed due to cleanup

Even McDonald's on W. 125th & Broadway was closed

Most city parks remain closed

Southbound Riverside Drive, around W. 116th St: with southbound Henry Hudson closed, cars jam Riverside Drive

Frick Collection, Frisk House, Frick Mansion

See previous related post: Alpine, NJ 07620: Nation's Most Expensive Zip Code

The Frick Collection is an art museum located on Manhattan's Upper East Side, at the corner of 5th Avenue and East 70th Street. The Frick House houses the Frick Collection today. The Frick Mansion is the 30,000 square-foot house, listed for $68 million in 2010, across the Hudson River in Alpine, New Jersey, country's most expensive zip code. All of these entities relate to Henry Clay Frick, "the Pittsburgh coke and steel industrialist," who lived from 1849 to 1919.

Frick was an art patron who collected paintings and other art objects. The Frick House was built in 1913 at its present location, and was built with Frick's intention that he would eventually "leave his house and his art collection to the public." The Frick Collection opened to public in 1935, 16 years after the death of Frick himself, and presently contains over 1000 works of art, "from the Renaissance to the late nineteenth century." The Frick Mansion in Alpine traces back to an estate the Frick family built in the 1930s. In 2006, real estate investor Richard Kurtz paid $58 million to acquire the 60-acre estate, and built the 30,000 square-foot mansion on the property. Immediately after completion, Kurtz put it on the market. The Frick Mansion, located on a completely gated drive, contains "12 bathrooms, 19 bedrooms, a library, a ballroom, a main kitchen, a catering kitchen, a basketball court, a movie theater and an 11-car garage," and furthermore can be controlled from anywhere with a smartphone.

Sources:

• http://quintessence823.blogspot.com/2012/04/alpine-nj-07620.html
• http://www.frick.org/collection/history.htm
• http://www.businessinsider.com/most-expensive-smart-home-alpine-2012-1?op=1
• http://blogs.wsj.com/developments/2010/06/03/alpine-mega-mansion-most-expensive-home-most-expensive-zip-code/

Basic MATLAB Demonstrations of Markov Chain and Limiting Probabilities

Consider the following 3-state Markov chain denoted p:

    0.1000    0.4000    0.5000
    0.3000    0.6000    0.1000
    0.7000    0.2000    0.1000

The limiting probabilities (stationary vector) denoted by π satisfies the equation π = π*p. While the steps aren't shown here, π is calculated to be:
    0.3269    0.4423    0.2308

In the long-run, 32.69% of the time will be spent in state 1, 44.23% in state 2, and 23.08% in the rest. In the subsequent codes, the stationary vector is denoted by s, since MATLAB cannot work with the symbol π as a variable name.

s*p
    0.3269    0.4423    0.2308
This demonstrates that π*p indeed equals the original π stationary vector.

s*p(:,1)
    0.3269
s*p(:,2)
    0.4423
s*p(:,3)
    0.2308
Each of the operations s*p(:,i) represents multiply the 1x3 π stationary vector by the 3x1 i-th column vector of p. Each of the operations yielded the i-th entry of π. This demonstrates the property that π_j = Σ π_i * p_ij, summed over i, where j represents the column number of the matrix p. That also translates to π * (j-th column of p). The physical interpretation of s*p(:,1) is the long-run proportion of transitions into state 1. That number equals s(1), which is the proportion of times spent in state 1. It makes sense that the proportion of transitions into a certain state is also the proportion of times spent in that same state.