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In [[mathematics]], an [[integer sequence]] is called a '''complete sequence''' if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.
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For example, the sequence of [[powers of two]] {1, 2, 4, 8, ...}, based on the [[binary numeral system]], is a complete sequence; given any natural number, we can choose the values corresponding to the 1 bits in its binary representation and sum them to obtain that number (e.g. 37 = 100101<sub>2</sub> = 1 + 4 + 32). This sequence is minimal, since no value can be removed from it without making some natural numbers impossible to represent. Simple examples of sequences that are not complete include:
 
* The [[even number]]s; since adding even numbers produces only even numbers, no odd number can be formed.
* Powers of three; no integer having a digit "2" in its ternary representation (2, 5, 6...) can be formed.
 
== Conditions for completeness ==
 
Without loss of generality, assume the sequence ''a''<sub>''n''</sub> is in nondecreasing order, and define the partial sums of ''a''<sub>''n''</sub> as:
 
:<math>s_n=\sum_{m=0}^n a_m</math>.
 
Then the conditions
 
:<math>a_0 = 1 \,</math>
:<math>s_{k-1} \ge a_k - 1 \, \forall \, k \ge 1</math>
 
are both necessary and sufficient for ''a''<sub>''n''</sub> to be a complete sequence.<ref name="Honsberger">Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985, pp.123-128.</ref>
 
A corollary to the above states that
 
:<math>a_0 = 1\,</math>
:<math>2a_k \ge a_{k+1} \, \forall \, k \ge 1</math>
 
are sufficient for ''a''<sub>''n''</sub> to be a complete sequence.<ref name="Honsberger"/>
 
However there are complete sequences that do not satisfy this corollary, see {{OEIS|A203074}}.
 
== Other complete sequences ==
 
Below is a list of the better known complete sequences.
 
* The sequence of the number 1 followed by the [[prime number]]s (studied by [[Subbayya Sivasankaranarayana Pillai|S. S. Pillai]]<ref>S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp. 159–167.</ref> and others); this follows from [[Bertrand's postulate]].<ref name="Honsberger"/>
* The [[Fibonacci numbers]], as well as the Fibonacci numbers with any one number removed.<ref name="Honsberger"/> This follows from the identity that the sum of the first ''n'' Fibonacci numbers is the (''n''&nbsp;+&nbsp;2)nd Fibonacci number minus 1 (see [[Fibonacci_numbers#Second_identity]]).
* All Fibonacci n-Step numbers,<ref>{{Mathworld|title=Fibonacci n-Step Number|urlname=Fibonaccin-StepNumber}}</ref> where n=2 gives the Fibonacci numbers above, n=3 gives the [[Generalizations of Fibonacci numbers|Tribonacci numbers]] etc.
* The [[Lazy caterer's sequence]] that gives the maximum number of partitions that a plane can be divided into, using n straight lines as dividers.
* All higher dimensions of the Lazy caterer's sequence that uses n hyperplanes (of dimension d-1) to maximally divide a space (of dimension d). {{OEIS|id=A216274}}
* The Cookie cutter's sequence that gives the maximum number of partitions that a plane can be divided into, using n circles as dividers.<ref>{{Mathworld|title=Plane Division by Circles|urlname=PlaneDivisionbyCircles}}</ref>
* All higher dimensions of the Cookie cutter's sequence that uses n hyperspherical surfaces (of dimension d-1) to maximally divide a space (of dimension d). {{OEIS|id=A059250}}
 
== Applications ==
 
Just as the powers of two form a complete sequence due to the binary numeral system, in fact any complete sequence can be used to encode integers as bit strings. The rightmost bit position is assigned to the first, smallest member of the sequence; the next rightmost to the next member; and so on. Bits set to 1 are included in the sum. These representations may not be unique.
 
===Fibonacci coding===
 
For example, in the '''Fibonacci arithmetic''' system, based on the Fibonacci sequence, the number 17 can be encoded in six different ways:
 
:110111 (F<sub>6</sub> + F<sub>5</sub> + F<sub>3</sub> + F<sub>2</sub> + F<sub>1</sub> = 8 + 5 + 2 + 1 + 1 = 17, maximal form)
:111001 (F<sub>6</sub> + F<sub>5</sub> + F<sub>4</sub> + F<sub>1</sub> = 8 + 5 + 3 + 1 = 17)
:111010 (F<sub>6</sub> + F<sub>5</sub> + F<sub>4</sub> + F<sub>2</sub> = 8 + 5 + 3 + 1 = 17)
:1000111 (F<sub>7</sub> + F<sub>3</sub> + F<sub>2</sub> + F<sub>1</sub> = 13 + 2 + 1 + 1 = 17)
:1001001 (F<sub>7</sub> + F<sub>4</sub> + F<sub>1</sub> = 13 + 3 + 1 = 17)
:1001010 (F<sub>7</sub> + F<sub>4</sub> + F<sub>2</sub> = 13 + 3 + 1 = 17, minimal form, as used in [[Fibonacci coding]])
:The maximal form above will always use F<sub>1</sub> and will always have a trailing one. The full coding without the trailing one can be found at
:{{OEIS|id=A104326}}. Note that by dropping the trailing one, the coding for 17 above occurs as the 16th term of A104326.
:The minimal form will never use F<sub>1</sub> and will always have a trailing zero. The full coding without the trailing zero can be found at
:{{OEIS|id=A014417}}. This coding is known as the [[Zeckendorf's theorem|Zeckendorf representation]] .
 
In this numeral system, any substring "100" can be replaced by "011" and vice versa due to the definition of the Fibonacci numbers.<ref>{{cite web |url=http://www.goldenmuseum.com/1202FibCdeTransf_engl.html |title=The main operations of the Fibonacci arithmetic |author=Alexey Stakhov |work=Museum of Harmony and Golden Section |accessdate=27 July 2010}}</ref>
 
===Other coding systems===
 
Other coding systems can be similarly devised. As with the Fibonacci sequence above, these coding systems that employ complete sequences will need to deal with multiple encoding solutions. The two main methods used are the [[greedy algorithm]] that will attempt to minimize the number of terms needed to encode an integer from the complete sequence and a minimizing algorithm that will attempt to maximize the number of terms needed to encode the same integer.
* A coding for the sequence of the number 1 followed by the prime numbers using the greedy algorithm can be found at
:{{OEIS|id=A007924}}.
* A coding for the sequence of the number 1 followed by the prime numbers using a minimizing algorithm can be found at
:{{OEIS|id=A205598}}.
* A coding for the Lazy caterer's sequence using the greedy algorithm can be found at
:{{OEIS|id=A204009}}.
 
==See also==
* [[Ostrowski numeration]]
 
== External links ==
 
* {{MathWorld | urlname=CompleteSequence | title=Complete Sequence}}
 
== References ==
<references/>
 
[[Category:Integer sequences]]

Latest revision as of 18:58, 27 December 2014

Every of the new knives feature a stainless-steel blade with a sharp leading edge that the person can re-sharpen, and are all designed with specific needs and preferences in mind. With a curved "Hawk" Blade, the Fastback Hawk Invoice Knife is ideal for pull cuts or slicing across the edge of cable to take away set up. With a pointy, pointed tip, The Fastback Easy Pocket Knife is designed for controlled piercing and smooth common goal chopping. Lastly, the Fastback Spring Assisted Knife features a black oxide coated blade for ultimate durability and sharpness, and a serrated edge for tearing by way of material like rope.



A few of my most irritating experiences in presenting have been in larger institutional venues the place laws and demarcation rules prevent you from being self-reliant. Within the absence of the authorised particular person, your presentation can fail on account of bureaucratic processes that depart everyone disempowered. On more than one occasion I've gerber tools ignored rules and made minor bodily modification of methods to allow a presentation to proceed. Having my Leatherman readily available has been important. When establishing shows, especially at outside venues dealing with wind and weather, my Leatherman, some blue-tack, cord and gaffer tape, have solved sudden issues on many occasions.

A stop over at the auto elements retailer had me finishing up a quart dimension can of bondo, a second tube of cream harder, a selection of tough and fine sandpaper and a twig can of matching paint. Again at the house I set every thing out I would wish for this newest repair. I exploit an orbital sander for small jobs like these and I reduce the sandpaper up into the proper sizes to fit in the sander. I used a pointy pocket knife to carve a number of milk jugs up into helpful six by 9 inch items of plastic. And lastly, some plastic window screening materials.

You'll have no problem find a knife that's utilized for a basic objective or a knife that has a historical worth or a elaborate collectible batman batarang and tac force knives. There's all the time knife obtainable that can fulfill your aesthetic wants. One will be concerned with buying a knife with a plain deal with or a knife with a complicatedly fashioned deal with, and he'll have the ability to own one with none hassle. These Batman batarang is double bladed. They are fabricated from stainless steel and utilizes aluminum made deal with. The blades are sharp they usually a special emblem on the handle.

The history behind the Mental Hospital is the place the "rumors" start. In response to city documents, the construction first served as a sanatorium to deal with tuberculosis and polio, however years later it was converted into a psychiatric asylum, where many lives had been misplaced because of the ineffectiveness of drugs and outdated strategies to treat the insane; the lives of those who needed to be cured and recurred to the hospital with hope, however never made it out, at least with a smile. Within the 1980's the constructing was shut down and changed by the newly remodeled Rancho Los Amigos Hospital.

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Weapons and Knives are a must, the more the higher. The weapons actually don’t even should shoot at all. Maybe some of them, but for essentially the most part they’re just for intimidation. Identical concept for the knives. You should have knives strapped to yourself in every single best Tactical knife review place. Huge knives. None of those tine ten dollar lock blade pocket knives from the boy scouts. They should be large and conspicuous. It is crucial that they are visible so that everyone can see them. Their additionally useful for grooming your hair or scratching stick figures onto the table on the local bar.

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