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.

Jar

A jar is a rigid, approximately cylindrical container with a wide mouth or opening. Jars are typically made of glass, ceramic, or plastic. They are used for foods, cosmetics, medications, and chemicals that are relatively thick or viscous: pourable liquids are more often packaged in a bottle. They are also used for items too large to be removed from a narrow neck bottle.

Glass jars can be used for home canning and food preservation. They can be used to preserve or store items as diverse as jam, pickled gherkin, other pickles, marmalade, sundried tomatoes, olives, jalapeño peppers, chutneys, pickled eggs, honey, and many others. They are also frequently re-used in order to put home-made preserves in. Jars are sterilised by putting them in a pressure cooker with boiling water or an oven for a number of minutes. If they are not required for further storage of items, they can be recycled.

A closure applied to the mouth of a jar can be a screw cap, lug cap, cork stopper, or other suitable means.

JAR (file format)

In software, JAR (Java Archive) is a package file format typically used to aggregate many Java class files and associated metadata and resources (text, images, etc.) into one file to distribute application software or libraries on the Java platform.

JAR files are fundamentally archive files, built on the ZIP file format and have the .jar file extension. Computer users can create or extract JAR files using the jar command that comes with a JDK. They can also use zip tools to do so; however, the order of entries in the zip file headers is important when compressing, as the manifest often needs to be first. Inside a JAR, file names are unicode text.

Design

A JAR file allows Java runtimes to efficiently deploy a set of classes and their associated resources. The elements in a JAR file can be compressed, which, together with the ability to download an entire application in a single request, makes downloading a JAR file much more convenient than separately downloading the many uncompressed files which would form a single Java Application. The package java.util.zip contains classes that read and write JAR files.

Jar (album)

Jar is the debut studio album by American band Superheaven, released on April 30, 2013. It was originally released under the band's prior name, Daylight, which has since been changed due to a legal dispute.

Track listing

All music composed by Superheaven.

Personnel

References