Virtually, any uniquely decodable one-to-many code, not necessary a prefix one, must satisfy Kraft's inequality.

This is an infinite sum which has one summand for every p in the domain of F. The requirement that the domain be prefix-free, together with Kraft's inequality, ensures that this sum converges to a real number between 0 and 1.

In coding theory, Kraft's inequality, named after Leon Kraft, gives a sufficient condition for the existence of a prefix code and necessary condition for the existence of a uniquely decodable code for a given set of codeword lengths.

More specifically, Kraft's inequality limits the lengths of codewords in a prefix code: if one takes an exponential function of each length, the resulting values must look like a probability mass function.

Kraft's inequality can be thought of in terms of a constrained budget to be spent on codewords, with shorter codewords being more expensive.

* If Kraft's inequality holds with strict inequality, the code has some redundancy.

* If Kraft's inequality holds with strict equality, the code in question is a complete code.

* If Kraft's inequality does not hold, the code is not uniquely decodable.

Kraft's inequality was published by.

However, Kraft's paper discusses only prefix codes, and attributes the analysis leading to the inequality to Raymond Redheffer.

Kraft's inequality states that

Hence the bit strings are prefix codes, and Kraft's inequality gives that.

Kraft's inequality therefore also holds for any prefix code.

* Kraft's inequality ( NIST )

When sold in Europe, such cheese are obliged to use another name, such as Kraft's " pamesello italiano ".

The media took up the search on Kraft's behalf with all eight children identified in eight days and resulted in many TV specials and interviews in the Australian National media.

* On September 24, 2012, UnitedHealth Group replaced Kraft Foods following Kraft's spinning off its North American grocery business.

Starting with Swamp Thing Annual # 6, Collins moved on to write Swamp Thing # 110-138, dramatically overhauling the series by restoring the pre-Alan Moore tone and incorporating a new set of supporting cast members into the book .< ref >" Nancy Collins: Swamp Thing < nowiki >'</ nowiki > s New Scripter Speaks ," David Anthony Kraft's Comics Interview, # 102 ( 1991 ), pp. 4-13 .</ ref > Collins resurrected Anton Arcane along with the Sunderland Corporation as foils for Swamp Thing.

There, Wirtanen makes an appearance to warn Campbell of Kraft's plot, and of Resi's complicity in it.

During the Second World War, rationing of meat and dairy products lead to Kraft's wide popularity.

April 2, 2007 marked Kraft's first day trading as an independent company on the New York Stock Exchange.

Kraft's parents moved to California from Wyoming prior to his birth.

The Kraft's new family home in Midway City, the only one on the block at that time, was a small, wood-frame Women's Army Corps dormitory on Beach Boulevard that Kraft's father Harold turned into a three-bedroom house.

Kraft's mother Opal served on the PTA at Kraft's school, Midway City Elementary, and baked cookies for Kraft's Cub Scout meetings.

Beach Boulevard was expanded from two-lanes to four-lanes in 1954, when Kraft was 9, and the Kraft home and chicken farm eventually was sandwiched between a blacksmith shop and the Red Garter cocktail lounge, whose patrons often left used condoms in Kraft's backyard that Kraft was tasked to shovel up every Sunday morning after church.

Dukette is thought to be Kraft's first victim.

To date, 22 of Kraft's 67 estimated victims remain unrecovered and unidentified.

Certain details surrounding some of Kraft's murders have caused many to suspect that Kraft did not always act alone.

* Forensic evidence in two cases point to an accomplice — an extra set of footprints and semen that did not match Kraft's DNA.

* The snapshots Kraft had of the dead men were processed somewhere, but no photo developer reported Kraft's morbid images to the police.

Graves died of AIDS before police could question him, so the question of Kraft's accomplice was never raised in court.

) Jackson also told McDougal that the list included only Kraft's " more memorable " murders, saying the total body count stood closer to 100.

Entering a sexual relationship with a subordinate, even when the contact is initiated by the latter, is unethical because of the subordinate's vulnerability to the physician and the inequality of power that characterizes the relationship.

A study published in 2009 took into account data sets from major world economies and correlated them with inequality indices.

There are some problems in both NP and co-NP, for example all problems in P or integer factorization, however it is not known if the sets are equal, although inequality is thought more likely.

In combinatorial mathematics, the Lubell – Yamamoto – Meshalkin inequality, more commonly known as the LYM inequality, is an inequality on the sizes of sets in a Sperner family, proved by,,, and.

All these inequalities, like the original inequality of Bell, by assuming local realism, place restrictions on the statistical results of experiments on sets of particles that have taken part in an interaction and then separated.

One is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval.

In other words, every block code satisfies the inequality.

Conversely, for a given set of natural numbers satisfying the above inequality, there exists a uniquely decodable code over an alphabet of size with those codeword lengths.

satisfying the Kraft inequality, one can construct a prefix code with codeword lengths equal to by pruning subtrees from a full-ary tree of depth.