| 1. |
- Eriksson, Kimmo, 1967-, et al.
(författare)
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An Exponential Limit Shape of Random q-proportion Bulgarian Solitaire
- 2018
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Ingår i: Integers. - 1553-1732. ; 18
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Tidskriftsartikel (refereegranskat)abstract
- We introduce pn-random qn-proportion Bulgarian solitaire (0 < pn, qn ≤ 1), playedon n cards distributed in piles. In each pile, a number of cards equal to the propor-tion qn of the pile size rounded upward to the nearest integer are candidates to bepicked. Each candidate card is picked with probability pn, independently of othercandidate cards. This generalizes Popov’s random Bulgarian solitaire, in whichthere is a single candidate card in each pile. Popov showed that a triangular limitshape is obtained for a fixed p as n tends to infinity. Here we let both pn and qnvary with n. We show that under the conditions q2npnn/log n → ∞ and pnqn → 0 asn → ∞, the pn-random qn-proportion Bulgarian solitaire has an exponential limitshape.
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| 2. |
- Krüger, Oliver, 1988-
(författare)
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Note on Odd/Odd Vertex Removal Games on Bipartite Graphs
- 2014
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Ingår i: Integers. - 1553-1732. ; 14, s. 1-4
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Tidskriftsartikel (refereegranskat)abstract
- We analyze the Odd/odd vertex removal game introduced by P. Ottaway. We prove that every bipartite graph has Grundy value 0 or 1 only depending on the parity of the number of edges in the graph, which is a generalization of a conjecture of K. Shelton.
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| 3. |
- Kurlberg, Par, et al.
(författare)
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On Sets of Integers Which Are Both Sum-Free and Product-Free
- 2013
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Ingår i: Integers. - 1553-1732.
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Tidskriftsartikel (refereegranskat)abstract
- We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.
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| 4. |
- Larsson, Urban, 1965
(författare)
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A Generalized Diagonal Wythoff Nim
- 2012
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Ingår i: Integers. - 1553-1732. ; 12:G2, s. 1-24
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Tidskriftsartikel (refereegranskat)abstract
- The P-positions of the 2-pile take-away game of Wythoff Nim lie on two beams of slope (sqrt(5)+1)/2 and (sqrt(5)−1)/2 respectively. We study extensions to this game where a player may also remove simultaneously pt tokens from either of the piles and qt from the other, where p < q are given positive integers and where t ranges over the positive integers. We prove that for certain pairs (p, q) the P-positions are identical to those of Wythoff Nim, but for (p, q) = (1, 2) they do not even lie on two beams. By several experimental results, we conjecture a classification of all pairs (p, q) for which Wythoff Nim’s beams of P-positions transform via a certain splitting behavior, similar to that of going from 2-pile Nim to Wythoff Nim.
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| 5. |
- Larsson, Urban, 1965
(författare)
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A Generalized Diagonal Wythoff Nim
- 2012
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Ingår i: Integers. - 1553-1732. ; 12:G2, s. 1-24
-
Tidskriftsartikel (refereegranskat)abstract
- The P-positions of the 2-pile take-away game of Wythoff Nim lie on two beams of slope (sqrt(5)+1)/2 and (sqrt(5)−1)/2 respectively. We study extensions to this game where aplayer may also remove simultaneously pt tokens from either of the piles and qt from the other, where p
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| 6. |
- Merikoski, Jorma K., et al.
(författare)
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Complete Additivity, Complete Multiplicativity, and Leibniz-additivity on Rationals
- 2021
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Ingår i: Integers. - : Colgate University. - 1553-1732. ; 21
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Tidskriftsartikel (refereegranskat)abstract
- Completely additive (c-additive in short) functions and completely multiplicative (c-multiplicative in short) functions are ordinarily defined for positive integers but sometimes on larger domains. We survey this matter by extending these functions first to nonzero integers and thereafter to nonzero rationals. Then we can similarly extend Leibniz-additive (L-additive in short) functions. (A function is L-additive if it is a product of a c-additive and a c-multiplicative function.) We study some properties of these functions. The role of an L-additive function as a generalized arithmetic derivative is our special interest.
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| 7. |
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