great pictures

The nature of democracy

From Znet

The media therefore presents gossip not in spite of American democracy, but to enhance and preserve a certain conception of it: one that involves spectators, not participants; public ratification, not public decision making. It is antithetical to a participatory economy and the idea of self-governance, and displays a striking commitment to reactionary ideology, despite illusions of an independent press. The issue at hand is and will always be whether or not the media is free, but will remain unresolved as long as the media is responsible for the gossip that debases public life.

William E. Shaub is a violin performance major at the Juilliard School of Music in Manhattan and the editor of TheFBM.com.

 

How true. Yukie says that in Japan scandals are more about connections to unpopular groups or money scandals. At least that appears to have some political relevance...though it rarely does...

http://www.nytimes.com/2012/01/22/sunday-review/hard-truths-about-disclosure.html?hp

To illustrate how few people actually read its terms and conditions disclosure, the online retailer Gamestation, on April Fools’ Day 2010, replaced the usual text with what it called an “immortal soul clause,” which read: “By placing an order via this Web site on the first day of the fourth month of the year 2010 anno Domini, you agree to grant us a non-transferable option to claim, for now and forever more, your immortal soul.” Eager to get on with their online purchase, 88 percent of customers clicked the box to sell their souls. (The 12 percent who opted out were rewarded with a cash credit for their diligence.)

factorial functionSSSSS

Abstract. The Gamma function of Euler often is thought as the only function which interpolates the factorial numbers n! = 1,2,6,24,.... This is far from being true. We will discuss four factorial functions
the Euler factorial function n!,
the Hadamard Gamma function H(n),
the logarithmic single inflected factorial function L(n),
the logarithmic single inflected hyper-factorial function L*(n).

We will show that these alternative factorial functions posses qualities which are missing from the the conventional factorial function but might be desirable in some context.

how big of a mammal are we operating at?

rest state 100 watts

as a social animal...15,000

more than biggest mammal blue whale

A castle without a dragon is worse than no castle at all.

legos and boys

The idea of creative play as conducive to learning, or even formal education, is an article of faith at Lego that goes back to its founder, who defended his decision to become a toymaker during the Great Depression by pointing out that all animals use play to develop their brains. In Japan, however, Lego found that study and play were more clearly delineated. Few Japanese parents bought Lego, as they do in Germany or the U.S., because they were “toys with vitamins in them,” as Lego senior director Søren Holm only half-jokingly puts it.

American boys, meanwhile, turned out to be the least free of any group Lego tracked. British and German boys are far more likely to play unsupervised in yards and wooded areas and even have greater latitude in decorating their bedroom walls. Among slightly older American boys, 9 to 12, building with Lego represented a rare chance to be left alone. (On one subject, boys of all ages and nationalities agreed: A castle without a dragon is worse than no castle at all.)

 

One can construct the polars of a polynomial by means of a differential operator. Suppose we have a homogeneous polynomial p(x1xn) To compute the polars ofp we act on it with the operator =y1x1++ynxn the k th polar of p equals kp(x1xn)

Linearization is the process of reducing a homogeneous polynomial into a multilinear map over a commutative ring. There are in general two ways of doing this:

• Method 1. Given any homogeneous polynomial f of degree n in m indeterminates over a commutative scalar ring R (scalar simply means that the elements of Rcommute with the indeterminates).
  

  Step 1

  If all indeterminates are linear in f , then we are done.

  Step 2

  Otherwise, pick an indeterminate x such that x is not linear in f . Without loss of generality, write f=f(xX) , where X is the set of indeterminates in fexcluding x . Define g(x1x2X):=f(x1+x2X)−f(x1X)−f(x2X) . Then g is a homogeneous polynomial of degree n in m+1 indeterminates. However, the highest degree of x1x2 is n−1 , one less that of x .

  Step 3

  Repeat the process, starting with Step 1, for the homogeneous polynomial g . Continue until the set X of indeterminates is enlarged to one X such that eachxX is linear.



• Method 2. This method applies only to homogeneous polynomials that are also homogeneous in each indeterminate, when the other indeterminates are held constant, i.e., f(txX)=tnf(xX) for some n and any tR . Note that if all of the indeterminates in f commute with each other, then f is essentially amonomial. So this method is particularly useful when indeterminates are non-commuting. If this is the case, then we use the following algorithm:
  

  Step 1

  If x is not linear in f and that f(txX)=tnf(xX) , replace x with a formal linear combination of n indeterminates over R :

  r1x1++rnxn, where riR

  Step 2

  Define a polynomial gRx1xn , the non-commuting free algebra over R (generated by the non-commuting indeterminates xi ) by:

  g(x1xn):=f(r1x1++rnxn)

  Step 3

  Expand g and take the sum of the monomials in g whose coefficent is r1rn . The result is a linearization of f for the indeterminate x .

  Step 4

  Take the next non-linear indeterminate and start over (with Step 1). Repeat the process until f is completely linearized.

Remarks.

1. The method of linearization is used often in the studies of Lie algebras, Jordan algebras, PI-algebras and quadratic forms.
2. If the characteristic of scalar ring R is 0 and f is a monomial in one indeterminate, we can recover f back from its linearization by setting all of its indeterminates to a single indeterminate x and dividing the resulting polynomial by n! :
   f(x)=1n!linearization(f)(xx)
   Please see the first example below.
3. If f is a homogeneous polynomial of degree n , then the linearized f is a multilinear map in n indeterminates.

Examples.

• f(x)=x2 . Then f(x1+x2)−f(x1)−f(x2)=x1x2+x2x1 is a linearization of x2 . In general, if f(x)=xn , then the linearization of f is
  Snx(1)x(n)=Snni=1x(i) 
  where Sn is the symmetric group on 1n . If in addition all the indeterminates commute with each other and n!=0 in the ground ring, then the linearization becomes
  n!x1xn=ni=1ixi