Trivia - A Google a Day XXVI

Who was Stepan Oblonsky's sister and Vronsky's lover?
Anna Karenina

Who commissioned Michelangelo Buonarroti to paint the frescos on the ceiling of the Sistine Chapel in 1508?
Pope Julius II

Which is the first NFL team to have lost four Super Bowls in a row?
Buffalo Bills

In the U.S., some states are actually "commonwealths". Which of these reaches farthest west?
Kentucky

In formal portraits of the military commander defeated at Yorktown in 1781, what color is his jacket?
Red

In 1949, when the Communist Party of China took control of mainland China, to where did the Kuomintang retreat?
Taiwan

Contrary to popular belief, a 2010 study showed what animal sleeps only about 10 hours, as opposed to the 18-20 previously thought to be the case?
Sloth

What is the location of the confluence of the two rivers that create the largest tributary to the Mississippi River?
Pittsburgh, Pennsylvania

If you visited Ann-Margaret's star on the Hollywood Walk of Fame, then walked towards North Highland Avenue, what theater would you pass by a few blocks down?
El Capitan Theatre

What is the loudest thing in the ocean, which, at 246dB in water, is capable of whiting out a submarine's SONAR?
Shrimp Layer

Trivia - A Google a Day XXV

What weather phenomenon characterizes the local climate of Thailand?
Monsoons

The group once known as "The Elgins" is said to have been "as influential to soul as The Beatles are to pop and rock". To what did this group change its name in the early 1970s?
The Temptations

Who apart from Bill Bradley is the only player ever to win the Euroleague, an Olympic Gold Medal and an NBA Championship?
Manu Ginobili

What is Orwell's home country renamed under the Party's rule in his 1949 dystopian novel?
Airstrip One

What practice, which had the roots of opposition in the Declaration of Independence, was actually banned in the Northwest Territory by Northwest Ordinance of 1787?
Slavery

Which high official is duped by the "honest" villain, calling a trial in the middle of the night as a result, in the first act of Shakespeare's play about the African-born general?
Brabantio

What water pump, situated between Dufours Place and Cambridge Street, was the culprit for the spreading of disease in London's 1854 epidemic?
Broad Street Pump

What is the 3-letter code for the largest airport in the city where you will find Congo Square?
MSY

Which two-time winner of The Open Championship married a six-time winner of the US Tennis Open in 2008?
Greg Norman

In 2007, what team had one of the oldest players ever to play in the NBA?
Dallas Mavericks

Differential Equations - Separation of Variables

Solving a differential equation by separation of variables is usually the first thing that shows up in an introduction to differential equations. The basic idea is just to group the variables on one side or another, as in the following example. The main tricky part is just to remember the different ways in which a partial derivative can be written.

y'=dy/dx=y_x

Are all equivalent. Also, although its usually preferred to be written in a more descriptive way, d/dx is sometimes written as D. Now lets work through an example.

y' = -4x/y

y dy/dx = -4x

y dy + 4x dx = 0

y²/2 + 2x² = C

y² + 4x² = C

The reason I moved the 4x over to the left hand side is because since we know that the two will integrate to x^2 and y^2, the solution must be an ellipse with the size defined as the combination of the constants due to integration left on the right hand side. Thus if we have some initial values, such as

y(2)=3

Thus our constant will solve out to

C = 3² + 4(2)²

Giving us a solution of

C = 25
y² + 4x² = 25

Although basic there is a lot of use to separation of variables. Combining separation of variables and reasoning through the specified class of functions we know the system must behave like is taught as proofs for many of the equations in electrostatics.

Trivia - A Google a Day XXIV

Whose last novel won the Pulitzer Prize for Fiction in 1963?
William Faulkner

What was the relationship of the author of "Jane Eyre" and the author of "Wuthering Heights"?
Sisters

In images of George W. Bush with his dog, Barney, what color is the dog?
Black

What astronomy system prevailed until Copernicus' proposed model of the solar system was accepted?
Ptolemaic

What famous aviator and his family left the U.S. after being the victims of the "Crime of the Century" in 1932?
Charles Lindbergh

Besides the Milky Way, what other galaxy is visible to a resident of Lansing, Michigan?
Andromeda

On what river is the graduate school attended by the eldest son of the 41st POTUS?
Charles River

What is the most populated city of the Irish province that flies a blue flag that features three gold crowns?
Cork

What is the given first name of the former NFL quarterback who has three sons who played college football, two of whom went on to be quarterbacks in the NFL?
Elisha

Because its original epithet discouraged sailors, what was the name changed to by Portugal's King John II?
Cape Of Good Hope

Linear Algebra - Singular Value Decomposition

Singular value decomposition is the process of factorization for a real or complex matrix. The goal is to divide this one matrix into three separate matrices that combine to reform this original matrix. First we start with an arbitrary matrix.

Then we take the transpose of this matrix and multiply it against the first in an A^TA fashion.

giving us the following result

We then find the eigenvalues of this matrix by subtracting λI_n and then solve the characteristic equation.
The characteristic equation is found by taking the determinant of the adjusted matrix.

(40-λ)(85-λ)-(-30)(-30) = 0

3400 -125 λ + λ² - 900 = 0

λ² -125 λ + 2500 = 0

(λ-25)(λ-100)=0

λ = 25,100

We then take the square roots of these values in order to get the sigma values, ordering them from highest to lowest for our diagonal matrix S.

σ₁=10, σ₂=5

giving us an S matrix of

These values will also be used in creating our U matrix. But in order to find our U matrix we must first find our V matrix so we need to orthogonalize the eigenvectors from our A^TA matrix. Our first step will be to substitute in our eigenvectors in the same order we have our sigma values.

So solving for the kernel by setting this equal to 0 gives us the vector

Now we do the same for the other eigenvalue

So after solving for the kernel we will get the vector

Now we have to normalize these two vectors so we take their magnitude, and divide the vectors by this amount. 

((1²) + (2²))^1/2

(5)^1/2

Giving us the following V matrix.

To form the U matrix we use the following method to compose the u vectors.

u_n=Av_n/σ

u₁=Av₁/σ

u₂=Av₂/σ

Now we get the combined U matrix of

This gives us all the parts we need to find the svd decomposition, the only difference is that our V matrix is actual V transpose in the equation so we have the following matrixes

U=S=V=

And the equation to regain our original matrix is M = USV^T. You can check my multiplying these out to confirm that they give you the original matrix. Thus we have completed our singular value decomposition of our matrix!

U=S=V^T=

Trivia - A Google a Day XXIII

Who set a MLB record, surpassing Lou Gehrig, for playing in the most consecutive games before he retired in 1998?
Cal Ripken, Jr.

What is the given name of the youngest child of the "King of Pop"?
Prince Michael Jackson II

What period of U.S. history, lasting until 1877, followed the conclusion of the American Civil War?
Reconstruction

What former editor of Time Asia wrote a book about the rare coronavirus that became an epidemic in China during 2003?
Karl Taro Greenfeld

How tall is the "Advanced Step in Innovative Mobility" creation?
Four Feet

Apart from Michael Jordan and Patrick Ewing who was the only NBA player in the Gold Medal-winning squads at the 1984 and the 1992 Olympic Games?
Chris Mullin

When the krewe that is the "King of Mardi Gras" makes its way along its traditional parade route, onto what street does it turn when it comes to the end of St. Charles?
Canal Street

George Read represented Delaware at the Continental Congress. What position did he old at the time of his death?
Chief Justice of the State of Delaware

What should every being in the universe never forget to carry with them according to the alien researching Earth in Adams' 1979 novel?
Towel

Beginning in 1894, a Russian jeweler was commissioned annually to make a special gift for the czar to give to his wife. What was the surname of this jeweler?
Faberge

Trivia - A Google a Day XXII

Who, along with two friends, painted graffiti on buildings in Lower Manhattan, tagging each as the work of "SAMO"?
Jean-Michel Basquiat

What is the title of Eliot Asinof's book subtitled "The Black Sox and the 1919 World Series"?
Eight Men Out

What do thousands of people traditionally gather to celebrate each year at W. 46th Street and Broadway in New York City?
New Year's Eve

What role did a two-time Oscar-winning actress turn down because she thought her co-star was going to be Errol Flynn, with whom she refused to work?
Scarlett O'Hara

Since 2003, Egypt has demanded the return of what ancient artifact that's been on display in a Berlin museum since before WWII?
Bust Of Nefertiti

Despite the legend, it is not possible that Babe Ruth "called his shot" in the bottom of the ninth because the Yankees were away. In what venue did Ruth raise his right arm and point to center field?
Wrigley Field

In E.B. White's essay about his pig, who is the "dishonorable pallbearer"?
Fred

If you take Bus #919 from Deshengmen, travel to Donggou Village, Badaling Town, Yanqing County and purchase a ticket, to what would you gain entry?
Great Wall Of China

In August 1963, one of the biggest robberies in the history of Britain took place near Mentmore in Buckinghamshire. Considering the conversion rate at the time, what was the approximate value of the heist in U.S. dollars, rounded to the nearest million?
$7,000,000

This town, set on the Vineyard, is one America's earliest whaling towns and is home to one of the few nude beaches in the United States. What was this town's name before it was officially changed in 1998 to reflect its native American population?
Gay Head