Cage (graph theory)

In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth.

Formally, an (r,g)-graph is defined to be a graph in which each vertex has exactly r neighbors, and in which the shortest cycle has length exactly g. It is known that an (r,g)-graph exists for any combination of r ≥ 2 and g ≥ 3. An (r,g)-cage is an (r,g)-graph with the fewest possible number of vertices, among all (r,g)-graphs.

If a Moore graph exists with degree r and girth g, it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with odd girth g must have at least

vertices, and any cage with even girth g must have at least

vertices. Any (r,g)-graph with exactly this many vertices is by definition a Moore graph and therefore automatically a cage.

There may exist multiple cages for a given combination of r and g. For instance there are three nonisomorphic (3,10)-cages, each with 70 vertices : the Balaban 10-cage, the Harries graph and the Harries–Wong graph. But there is only one (3,11)-cage : the Balaban 11-cage (with 112 vertices).

Cage (enclosure)

A cage is an enclosure made of mesh, bars or wires, used to confine, contain or protect something or someone. A cage can serve many purposes, including keeping an animal in captivity, capturing, and being used for display of an animal at a zoo.

Humans

In history, prisoners were sometimes kept in a cage. They would sometimes be chained up inside into uncomfortable positions to intensify suffering.

History

Cages have been usually been used to capture or trapping a certain life form. For this reason, they've been known as a hunting accessory, often used for poaching animals or simply seizing them.

In captivity

Cages are often used now as a source to confine animals. These provide as a habitat to the animal, and since they've advanced so greatly, they are now specially designed to fit that species of animal. Captive breeds of birds, rodents, reptiles, and even larger animals have also been known to be confined in a cage as a domesticated animal (also known as a pet). Captivity is a common purpose of the cage.

Luke Cage

Luke Cage (born Carl Lucas and also called Power Man) is a fictional superhero appearing in American comic books published by Marvel Comics. Created by Archie Goodwin and John Romita, Sr., he first appeared in Luke Cage, Hero for Hire #1 (June 1972). Imprisoned for a crime he did not commit, he gains superpowers in the form of unbreakable skin and superhuman strength. The character frequently teams up with fellow superhero Iron Fist, and is married to Jessica Jones, with whom he has a daughter. In 2005, writer Brian Michael Bendis added Luke Cage to the lineup of the New Avengers, and he has appeared in various Avengers titles since.

Actor Mike Colter plays the character in Jessica Jones, a live-action television series set in the Marvel Cinematic Universe, and will headline in his own series, which will premiere in 2016.

Publication history

Luke Cage was created by Archie Goodwin and John Romita, Sr. shortly after Blaxploitation films emerged as a popular new genre. He debuted in his own series, Luke Cage, Hero for Hire, which was initially written by Goodwin and pencilled by George Tuska. Cage's adventures were set in a grungier, more crime-dominated New York City than that inhabited by other Marvel superheroes of the time. The series was retitled Luke Cage, Power Man with issue #17.

Greeks (finance)

In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are denoted by Greek letters (as are some other finance measures). Collectively these have also been called the risk sensitivities,risk measures or hedge parameters.

Use of the Greeks

The Greeks are vital tools in risk management. Each Greek measures the sensitivity of the value of a portfolio to a small change in a given underlying parameter, so that component risks may be treated in isolation, and the portfolio rebalanced accordingly to achieve a desired exposure; see for example delta hedging.

The Greeks in the Black–Scholes model are relatively easy to calculate, a desirable property of financial models, and are very useful for derivatives traders, especially those who seek to hedge their portfolios from adverse changes in market conditions. For this reason, those Greeks which are particularly useful for hedging—such as delta, theta, and vega—are well-defined for measuring changes in Price, Time and Volatility. Although rho is a primary input into the Black–Scholes model, the overall impact on the value of an option corresponding to changes in the risk-free interest rate is generally insignificant and therefore higher-order derivatives involving the risk-free interest rate are not common.

Lotna

Lotna is a Polish war film released in 1959 and directed by Andrzej Wajda.

Overview

This highly symbolic movie is both the director's tribute to the long and glorious history of the Polish cavalry, as well as a more ambiguous portrait of the passing of an era. Wajda was the son of a Polish Cavalry officer who was murdered by the Soviets during the Katyn massacre.

The horse Lotna represents the entire Romantic tradition in culture, a tradition that had a huge influence in the course of Polish history and the formation of Polish literature. Lotna is Wajda's meditation on the historical breaking point that was 1939, as well as a reflection on the ending of an entire era for literature and culture in Poland and in Europe as a whole. Writing of the film, Wajda states that it "held great hopes for him, perhaps more than any other." Sadly, Wajda came to think of Lotna "a failure as a film."

The film remains highly controversial, as Wajda includes a mythical scene in which Polish horsemen suicidally charge a unit of German tanks, an event that never actually happened.

Speed (Montgomery Gentry song)

"Speed" is a song written by Jeffrey Steele and Chris Wallin, and recorded by American country music duo Montgomery Gentry. It was released in December 2002 as the second single from their album My Town. The title from the cover of this single borrows its font from Speed Racer.

"She Couldn't Change Me" was included as a B-side.

Music video

The music video was directed by Trey Fanjoy. A young man trades his old truck for a car with speed as his truck just brings back memories of his ex-lover. He buys the car, then he drives the car really fast, but as he keeps seeing the memory of his ex-lover on the road, he jumps out of the car, and then he heads out running into the field. The duo is performing the song in the middle of a two-lane road at a night time setting.

Chart positions

"Speed" debuted at #57 on the U.S. Billboard Hot Country Songs for the week of December 28, 2002.

Year-end charts

References

External links