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A billiard ball is a small, hard ball used in cue sports, such as carom billiards, pool, and snooker. The number, type, diameter, color, and pattern of the balls differ depending upon the specific game being played. Various particular ball properties such as hardness, friction coefficient, and resilience are important to accuracy.
Rotation games do not distinguish between solids and stripes, but rather use the numbering on the balls to determine which object ball must be pocketed. In other games such as straight pool neither type of marking is of any consequence.
In British-style eight-ball pool and its blackball variant, fifteen object balls are used, but fall into two unnumbered groups, the reds (or less commonly blues) and yellows, with a white cue ball, and black 8 ball.
Ball sets for snooker consist of twenty-two balls in total, arranged as a rack of 15 unmarked red balls, six colour balls placed at various predetermined spots on the table, and a white cue ball. The colour balls are sometimes numbered with their point values in the style of pool balls for the home market.
Various other games have their own variants of billiard balls. English billiards uses the same number of balls as carom billiards, but the same size as snooker balls, as the game is played on the same size table as snooker. Each player uses a separate cue ball, with modern English billiards sets using one white ball with red spots and the other being yellow with red spots.
Because the collisions between billiard balls are nearly elastic, and the balls roll on a surface that produces low rolling friction, their behavior is often used to illustrate Newton's laws of motion. Idealized, frictionless billiard balls are a staple of mathematical theorems and physics models, and figure in dynamical billiards, scattering theory, Lissajous knots, billiard ball computing, and reversible cellular automata, Polchinski's paradox, contact dynamics, collision detection, the illumination problem, atomic ultracooling, quantum mirages, and elsewhere in these fields.
On the online encyclopedia Wikipedia are edited the articles Arithmetic billiards and Dedekind psi function, the online encyclopedia Wolfram MathWorld has edited the article Goldbach Conjecture. We consider the following examples as illustration of these images.
Figure F 1 represents the prime $2$ inside a box, as well as the figure F 2 is a representation for the prime number $3$ with two triggers that are the symbols $\circ$ for the representation of the entry and exit for a piece. For the primes $p\geq 3$ in these figures the greeen arrow represents the distance $D=\frac\varphi(p)2$, with $\varphi(n)$ the Euler's totient function. The box is a rectangle.
Example 2. I represent with a $\bullet$ a joint/bridge between two parts or pieces in our billiards. In this way the figure F 5 represents an application of Goldbach conjecture $16=\texteven integer=\textprime p+\textprime q=5+11$ with two triggers $\circ$ and a bridge $\bullet$. The distance to the wall is $D=-1+\frac\varphi(p)+\varphi(q)2$, therefore you can to think in the box as a rectangle.
Example 4. The figures F 9, F 10 and F 11 are for examples of billiards$\textodd prime number+\texteven integer,\tag1$The represented odd prime number is $5$ and the even integers are respectively $10,32$ and $16$. The expression $\textodd integer =\text odd prime number + \text even integer $ is by applciation of Goldbach conjecture, for odd integers $n\geq 7$.
Question. I would like to study the general case illustrated in figure F 11 (see also figures F 9, F 10 and F 12), in a similar way that the examples below. Can you deduce the (shape/dimensions of) box and the path defining an arithmetic billiard for the union (of the corresponding representations) of a general even integer, represented with the generic rectangle with orange colour and a general odd prime number, represented as the shape L in yellow, that's L is the representation of the odd prime $p=1+\varphi(p)$? Here $\varphi(n)$ is denoting the Euler's totient function. Also is required (if possible) add the formulas that you can to deduce defining your billiard, in a similar way that I did with the distance $D$. Many thanks.
In other words deduce a valid arithmetic billiard of a closed trajectory (a loop as in figures F 9, F 10 and F 11) for the (representation of) union of an even integer, represented by a rectangle $2\times n$ with $n>2$ and any odd prime number with the shape of a L, as in figures.
The following paragraphs are to illustate how to deduce more statments from these interesting (I think) billiards, for example the following is for a prime represented with the shape L, $p=1+\varphi(p)$, and we evoke a similar figure likes than the figure F8, for example the prime $p=17$, then one can to deduce a statement involving the prime $p$ the area of its representation L and the area of the box denoted as $V$. I hope that there aren't typos.
Figure F 1 illustrates the case of an even integer (in orange colour) represented by a rectangle of $2\cdot k$ rows and an even number of columns, while that in the case illustrated with figure F 2 the number of such columns is an odd integer $\geq 1$. In both cases the blue arrow equals to $D=\frac2k+p-12$ and the segments $H=D+2$ and $T=2k$ our even integer that share both cases (figures F1 and F2). You can observe the location of the exit of these billiards coloured with orange.
Figure F3 illustrates an example for the case with the exit of the billiard placed at the upper left corner of the orange rectangle that represents our even integer, in this case both rows and columns are even integers, our parameter $T$ is the number of rows, $T=2\cdot k$. The formulas are for the diagonal separation $S=\frac12\sqrt2$ coloured with green, the distance $D=\frac-12+\fracp-1+2k2$ and the height $H=D+2$.
Figure F5 illustrates one of the three cases in which the exit of the billiard that represents our even integer (the billiard coloured with orange colour) has the exit placed at the bottom left of such rectangle with an even number of columns and an even number of rows, the parameter $T=2\cdot k$. The three cases that we study for this billiard are $p-1=2k$ with $S'=\frac12\sqrt2$, $D=2k+2$ and $H=D-1$; the case $p-1
Figure F 6 is an example to illustrate one of the three cases in this, again the location of the exit of the billiard that represents our even integer is the same, but the number of rows is an odd integer. The three cases are for our parameter $T=2k$ that express the number of columns, $p-1=2k$ with $S'=1\cdot\sqrt2$, $D=2k+2$ and $H=D-1$; the case $p-1>2k$ has $S'=\sqrt2\fracp-1-22$, $D=H+1$ and $H=\fracp-1+2k+22$ and finally the case $p-1
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Up to about 1990 the quantum mechanics of classically chaotic systems,shortly termed "quantum chaos", was essentially a domain of theory(Haake 2001). Only two classes of experimental result had been available atthat time. First, there were the spectra of compound nuclei giving rise tothe development of random matrix theory in the sixties of the last century, andsecond the experiments with highly excited hydrogen andalkali atoms in strong magnetic or strong radiofrequency fields. The situation changed with the appearance of experiments using classical waves, starting with microwave billiards . The distinction between classical waves and matter waves is not of relevance in the present context, since all features touched in this article arecommon to all types of waves.This is why some authors prefer the term "wave chaos" to describe this field ofresearch.
In quantum mechanics this distinction between integrable and chaoticsystems does not work any longer. The initial conditions are defined only within the limits of the uncertaintyrelation\[ \Delta x\,\Delta p\ge \frac12\hbar\,,\]and the concept of trajectories looses its significance. One may even askwhether quantum chaos does exist at all. Since the Schrödinger equation islinear, a quantum mechanical wave packet can be constructed from theeigenfunctions by the superposition principle. There is no room left for chaos.On the other hand the correspondence principle demands that there must be arelation between linear quantum mechanics and nonlinear classical mechanics at leastin the regime of large quantum numbers. This defines the program of quantumchaos research, namely to look for the fingerprints of classical chaos in the quantummechanical properties of the system.
The first experiment of this type dates back already more then 200 years. At the end of the 18th century E. Chladni developed a technique "to make sound visible" by decorating the nodal lines of vibrating plates with grains of sand (Chladni 1802). Figure 2 shows Chladni figures for three typical situations. The plates are fixed in the centre and had been excited to vibrations by means of a loudspeaker. Figure 2(a) shows a typical pattern for a circular plate. In this case the integrability of the system is not perturbed by the mounting. One observes a regular pattern of nodal lines with many intersections. The next example in Figure 2(b) shows a rectangle. It is integrable, too, but now the integrability is slightly perturbed by the mounting resulting in a curvature of the nodal lines and a partial conversion of crossings into anti-crossings. The last example in Figure 2(c) belongs to the class of Sinai billiards, a rectangle with an excised quarter circle from one of the corners. Now all nodal line crossings have completely disappeared resulting in a meandering pattern of nodal lines. Two centuries after Chladni's discovery the study of nodal lines of chaotic plates has become a very active field of research again.
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