80,846 research outputs found

### Blocking Wythoff Nim

The 2-player impartial game of Wythoff Nim is played on two piles of tokens.
A move consists in removing any number of tokens from precisely one of the
piles or the same number of tokens from both piles. The winner is the player
who removes the last token. We study this game with a blocking maneuver, that
is, for each move, before the next player moves the previous player may declare
at most a predetermined number, $k - 1 \ge 0$, of the options as forbidden.
When the next player has moved, any blocking maneuver is forgotten and does not
have any further impact on the game. We resolve the winning strategy of this
game for $k = 2$ and $k = 3$ and, supported by computer simulations, state
conjectures of the asymptotic `behavior' of the $P$-positions for the
respective games when $4 \le k \le 20$.Comment: 14 pages, 1 Figur

### Trygg i transport â effekten av tid och upprepning pĂ„ unghĂ€stars lasttrĂ€ning

HĂ€star transporteras Ă¶ver hela vĂ€rlden i syften som fĂ¶rsĂ€ljning, tĂ€vling, trĂ€ning och djursjukhusvistelse. I mĂ„nga fall, men kanske frĂ€mst vid transport till djursjukhus, hinner aldrig den unga hĂ€sten fĂ„ trĂ€na sig pĂ„ att lastas innan den ska Ă„ka ivĂ€g nĂ„gonstans. Den unga hĂ€sten utsĂ€tts fĂ¶r mĂ„nga obekanta och frĂ€mmande situationer i samband med lastning, nĂ„got som medfĂ¶r att den blir stressad. Den obehagskĂ€nsla eller stress som hĂ€sten upplever kan uttryckas pĂ„ flera olika sĂ€tt, exempelvis beteendemĂ€ssigt eller fysiologiskt. Det krĂ€vs dĂ€rfĂ¶r att hĂ€sten lĂ€r sig vad transporten innebĂ€r och att det inte Ă€r nĂ„got att vara rĂ€dd fĂ¶r, vilket kan innebĂ€ra en svĂ„r uppgift fĂ¶r djurhĂ„llaren.
Syftet med den hĂ€r experimentella studien var att fĂ„ en uppfattning fĂ¶r hur den unga hĂ€sten pĂ„verkas av att lastas samt att ge en fĂ¶rstĂ„else fĂ¶r hur den orutinerade hĂ€stens inlĂ€rning kan tillĂ€mpas i vardagliga situationer dĂ„ hĂ€sthĂ„llare ska fĂ¶rbereda transport av sina unga hĂ€star. Studien inriktade sig dĂ€rfĂ¶r pĂ„ att ta reda pĂ„ hur hĂ€sten beter sig vid lastning, hur hĂ€stens puls fĂ¶rĂ€ndras under lastningstillfĂ€llet, hur lĂ„ng tid det tar att lasta en unghĂ€st och hur dessa faktorer fĂ¶rĂ€ndras med antalet gĂ„nger som hĂ€sten lastas. I studien anvĂ€ndes sex stycken hĂ€star av rasen islandshĂ€st. Samtliga hĂ€star lastades tre gĂ„nger vardera, en gĂ„ng om dagen i tre dagar. Beteenden som registrerades var fekaliserar, stĂ„r stilla och drar Ă„t sidan, det kunde utlĂ€sas att fler beteenden registrerades under dag 1 Ă€n under dag 2 och 3. Pulsen Ă¶kade markant nĂ€r hĂ€starna gick in i transporten jĂ€mfĂ¶rt med pulsen som mĂ€ttes innan lastning. Pulsen sjĂ¶nk betydligt nĂ€r hĂ€starna hade lastats av och stod Ă„ter pĂ„ fast mark igen. Tiden som det tog att lasta hĂ€starna minskade fĂ¶r varje dag som de lastades.
Studien visar att unghĂ€starna rent fysiologiskt Ă€r mycket mer pĂ„verkade av att stĂ„ inne i transporten Ă€n vad de Ă€r nĂ€r de stĂ„r pĂ„ vanlig fast mark. Studien klargĂ¶r ocksĂ„ att den unga hĂ€sten med hjĂ€lp av inlĂ€rning fĂ¶rstĂ„r vad det innebĂ€r att lastas dĂ„ resultatet tydliggĂ¶r en signifikant lĂ€gre puls men Ă€ven mindre uppvisade beteenden med antalet gĂ„nger som hĂ€sten blir lastad. Undantaget fĂ¶r den hĂ€r studien Ă€r dock hĂ€starna hade hĂ¶gre puls vid andra lastningstillfĂ€llet Ă€n vid fĂ¶rsta. Resultatet frĂ„n lastningstiden tolkas som att det gĂ„r fortare att lasta en hĂ€st pĂ„ en transport med antalet gĂ„nger som den blir lastad, Ă„tminstone om inget har skrĂ€mt hĂ€sten under tidigare lastningar. Att lastningstrĂ€na den unga hĂ€sten innan transportering anses vara av stort vĂ€rde fĂ¶r att fĂ¶rbĂ€ttra den unga orutinerade hĂ€stens vĂ€lfĂ€rd vid transport.Horses are transported around the world for purposes such as sales, competition, training and veterinary care. In many cases, mainly in transport to the veterinary hospital, the horse has not been trained to be loaded before being transported. The young horse is exposed to many unfamiliar and potentially frightening situations during loading, which can result in stress. The discomfort or stress the horse is experiencing can be expressed behaviorally and/or physiologically. It is therefore required that the horse is habituated to the vehicle and transportation, which could be a difficult task for the horse owner.
The aim of this study was to achieve an understanding of how the young horse is affected by loading. I also wanted to show how the inexperienced horse learning theory can be applied in everyday situations when horse keepers prepare for transporting their young horses. The study focused therefore on how the horse behaves when loading, how the horse's heart rate changes during the time of loading, how long it takes to load a young horse, and how these factors change with the number of times the horse is loaded. The study used six Icelandic horses. All horses were loaded three times each, once a day for three consecutive days. The results of the heart rate and loading time were compared and tested for statistical significance. Behaviors recorded were defecation, stand calm and pull to one side. It could be deduced that more behaviors were recorded on day 1 than on day 2 and 3. Heart rate was significantly higher when the horses were in the trailer (P= 0.042) compared with the heart rate that was measured before loading. Heart rate decreased significantly when the horses were unloaded and stood back on solid ground again (P= 0.008). The time it took to load the horses decreased significantly by day (P= 0.002).
The study shows that young horses, physiologically, is significantly affected by standing inside the vehicle compared to when they are outside on solid ground. The study also clarifies that the young horse with the help of learning, through repeated exposure, understands what it means to be loaded as the result elucidates a significantly lower heart rate, but also less behavior exhibited by the number of times that the horse will be loaded. Somewhat surprising was that the horses had a higher heart rate the second time of loading than the first. Loading time was reduced with an increasing number of times the horse was loaded. However, this effect might be reversed if something scares the horse during loading. To train the young horse of loading before transportation is considered to be of great value to improve the young horse welfare and human safety during transport

### The $\star$-operator and Invariant Subtraction Games

We study 2-player impartial games, so called \emph{invariant subtraction
games}, of the type, given a set of allowed moves the players take turn in
moving one single piece on a large Chess board towards the position
$\boldsymbol 0$. Here, invariance means that each allowed move is available
inside the whole board. Then we define a new game, $\star$ of the old game, by
taking the $P$-positions, except $\boldsymbol 0$, as moves in the new game. One
such game is \W^\star= (Wythoff Nim)$^\star$, where the moves are defined by
complementary Beatty sequences with irrational moduli. Here we give a
polynomial time algorithm for infinitely many $P$-positions of \W^\star. A
repeated application of $\star$ turns out to give especially nice properties
for a certain subfamily of the invariant subtraction games, the
\emph{permutation games}, which we introduce here. We also introduce the family
of \emph{ornament games}, whose $P$-positions define complementary Beatty
sequences with rational moduli---hence related to A. S. Fraenkel's `variant'
Rat- and Mouse games---and give closed forms for the moves of such games. We
also prove that ($k$-pile Nim)$^{\star\star}$ = $k$-pile Nim.Comment: 30 pages, 5 figure

### A Generalized Diagonal Wythoff Nim

In this paper we study a family of 2-pile Take Away games, that we denote by
Generalized Diagonal Wythoff Nim (GDWN). The story begins with 2-pile Nim whose
sets of options and $P$-positions are $\{\{0,t\}\mid t\in \N\}$ and
\{(t,t)\mid t\in \M \} respectively. If we to 2-pile Nim adjoin the
main-\emph{diagonal} $\{(t,t)\mid t\in \N\}$ as options, the new game is
Wythoff Nim. It is well-known that the $P$-positions of this game lie on two
'beams' originating at the origin with slopes $\Phi= \frac{1+\sqrt{5}}{2}>1$
and $\frac{1}{\Phi} < 1$. Hence one may think of this as if, in the process of
going from Nim to Wythoff Nim, the set of $P$-positions has \emph{split} and
landed some distance off the main diagonal. This geometrical observation has
motivated us to ask the following intuitive question. Does this splitting of
the set of $P$-positions continue in some meaningful way if we, to the game of
Wythoff Nim, adjoin some new \emph{generalized diagonal} move, that is a move
of the form $\{pt, qt\}$, where $0 < p < q$ are fixed positive integers and $t
> 0$? Does the answer perhaps depend on the specific values of $p$ and $q$? We
state three conjectures of which the weakest form is: $\lim_{t\in
\N}\frac{b_t}{a_t}$ exists, and equals $\Phi$, if and only if $(p, q)$ is a
certain \emph{non-splitting pair}, and where $\{\{a_t, b_t\}\}$ represents the
set of $P$-positions of the new game. Then we prove this conjecture for the
special case $(p,q) = (1,2)$ (a \emph{splitting pair}). We prove the other
direction whenever $q / p < \Phi$. In the Appendix, a variety of experimental
data is included, aiming to point out some directions for future work on GDWN
games.Comment: 38 pages, 34 figure

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