GYutsis 1.1

The GYutsis program has 2 modes: normal mode and advanced mode. When it initially starts it is always in normal mode.

GYutsis: normal mode

Defining a problem and generating the summation formula

All features described above can also be selected from the menus.

Changing the heuristic of the algorithm

Edge Cost Heuristic

this is the most simple and fastest heuristic: a cost is associated with each edge, equal to the difference in length of the two smallest cycles in which the edge participates. The cost of a cycle is defined as the minimum cost of its edges. This heuristic will interchange the edge with minimum cost out of the cycle with minimum cost. When two cycles have the same cost, the cycle for which the minimum edge cost is most reached is preferred, and if this is equal, the cycle with minimum total cost is choosen.

For small problems (upto 22 j's) this heuristic suffices, for higher problems the heavier heuristics provide shorter formulae.

More Smaller/Less Bigger Heuristic

this is the default heuristic and for most cases the best choice. This heuristic considers all possible interchanges making a girth cycle smaller. An interchange is preferred over an other if it makes more cycles of length l smaller, or if equal, makes less bigger. This criterium is repeatedly used for rising l starting at the girth until a difference found.

Cycle Count Heuristic

this heuristic also considers all possible interchanges making a girth cycle smaller, but prefers an interchange over an other if it it results in a graph with more cycles of length l. Again this criterium is used for rising l, starting at the girth minus 1, until a difference is found.

This heuristic yields shorter formulae than the Edge Cost Heuristic. For some cases it also performs better than the More Smaller/Less Bigger Heuristic, but these cases are rare.

Changing the format of the formula

Generic

a compact, human readable format. This is the default.

LaTeX

for the popular typesetting system LaTeX. By default a macro is used to represent the Wigner 6-j symbol. This can be turned of by clicking on the "Use macros" checkbox under the Output-menu. Afterwards the formula will be outputted in plain LaTeX.

Maple

for the popular computer algebra package Maple. Macros are used to represent the Kronecker delta symbol, the triangular symbol and the Wigner 6-j symbol (they are in fact Maple functions).

Racah

for the Maple package Racah (package especially for Racah algebra).

GYutsis: advanced mode

• The "User"-pane contains all the ouput generated by the user.
• In the "Operations"-pane all graph operations are logged. These operations can be:
  1. compound operations: removing/formatting a triangle, removing/formatting a bubble or formatting the graph as a triangular delta,
  2. elementary operations: performing an interchange, removing nodes, inverting node signs or inverting edge directions.
  By formatting we mean the process of locally inverting node signs and edge directions in order to get the subgraph in the correct form for applying a rule.
• In the "Rules"-pane all applied rules are logged from an algorithmic point of view: if we apply an interchange to reduce a square to a triangle, the square is shown as "Best Cycle" and the edge wich is interchanged out of the square as "best edge". From the graph operations point of view this will be an interchange on the "best edge". In addition equivalent operations (judged by the used heuristic) are also logged, if apparent.

The user operations are:

Print Graph

Outputs the graph in the following format (same example):

3
-0 | j1:-3 j2:-4 j5:+2 
-1 | j3:-3 j4:-4 j6:+2 
-2 | j5:-0 j6:-1 j7:+5 
+3 | j1:+0 j3:+1 j8:-5 
+4 | j2:+0 j4:+1 j9:-5 
+5 | j8:+3 j9:+4 j7:-2 

The first line contains the size of the problem n, for a general recoupling coefficient with n+1 leaves, corresponding with a cubic graph of 2n nodes and 3n edges. The next 2n lines contain the couplings of the graph:
<nodesign><node>| <edge> <edge> <edge>
with <edge>:= <label>:<direction><endpoint>, where direction is '+' for an incoming edge and '-' for a leaving edge.

Save Graph

Saves the graph graphically in gml-format for the graph drawing tool Graphlet. The signs of the nodes and the directions of the edges are ignored. A valid gml-file is generated with all the nodes on one (diagonally oriented) line. We recommend the graph drawing algorithm "General Graphs (Tunkelang)" for best results.

Graphlet is free of charge for non commercial use and can be downloaded from its homepage:
http://www.infosun.fmi.uni-passau.de/Graphlet/

Save each step

When this checkbox is selected, the graph will be saved graphically in gml-format after each step. The filenames will be of the form:

<filename>.k.gml

with <filename>.braket, <filename>.yts or <filename> the original filename of the Yutsis graph and k the step number.

When the braket is not saved yet, the filenames will be of the form:

newgraphx.k.gml

with x a number incremented with each anonymous graph and k the step number.

Cycles

Prints all the girth cycles of the graph.

Best Cycle

Prints the cycle the heuristic marks as the best cycle together with its best edge (if the cycle is bigger than a triangle). In addition equivalent operations are also printed, if apparent.

Step

Performs one step in the reduction process, i.e. applies one rule.

Print Macros

Prints default macros for the selected output format (only for LaTeX and Maple). In the case of Maple, these macros contain Maple functions for the Kronecker delta symbol, the triangular symbol and the Wigner 6-j symbol.

Each of the panes can be cleared by pressing the "clear"-button and the contents of a pane can be saved to a file by pressing the "Save to file"-button.

When the "Hide Advanced Panel"-button is pressed the program returns to normal mode without changing the state of the graph. All features present in normal mode are off course available in "advanced mode". Again all operations can be selected from the menus.

GYutsisApplet

CycleCostAlgorithm

 
Usage: java [-options] CycleCostAlgorithm <YutsisGraph> 
  <YutsisGraph>:= "<Bra|Ket>" | <filename>
  <filename> must be in BRAKET or YTS format
Options:
 heuristics:a: run both CycleCountHeuristics and output the best
            e: use the EdgeCostHeuristic
            b: use the CycleCountHeuristic.MORE_SMALLER_LESS_BIGGER (default)
            c: use the CycleCountHeuristic.CYCLE_COUNT
 output:    G: generic output (same as default but wrapped)
            L: LaTeX ouput
            M: Maple output
            R: Racah output
	    l: output macros for the LaTeX format
	    m: output macros for the Maple format
 general:   v: verbose output (same as "-or")
            o: output graph operations
            r: output rule selection
            g: ouput graph after each step in gml-format

The last argument can be a general recoupling coefficient in its mathematical standard format or a filename defining a general recoupling coefficient in BRAKET or YTS format.

The heuristic options determine the heuristic to be used:

a

Run the algorithm once with the More Smaller/Less Bigger Heuristic, once with the Cycle Count Heuristic and return the shortest formula.

e

The Edge Cost Heuristic will be used.

b

The More Smaller/Less Bigger Heuristic will be used.

c

The Cycle Count Heuristic will be used.

The output options define the format of the formula:

G

generic format. This is the same as the default but wrapped at 80 characters.

L

LaTeX format.

M

Maple format.

R

Racah format.

l

Output the default macros for the LaTeX format and exit.

m

Output the default macros for the Maple format and exit.

The general options define the output level of the program:

v

Verbose output: maximum output level; same as combination of both other options (or)

o

Output all operations performed on the graph. These operations can be compound operations: removing/formatting a triangle, removing/formatting a bubble or formatting the graph as a triangular delta; or elementary operations: removing nodes, inverting node signs or inverting edge directions. By formatting we mean the process of locally inverting node signs and edge directions in order to get the subgraph in the correct form for applying a rule.

r

Output the rules applied, from an algorithmic point of view: if we apply an interchange to reduce a square to a triangle, the square is shown as "Best Cycle" and the edge wich is interchanged out of the square as "best edge". From the graph operations point of view this will be an interchange on the "best edge". In addition equivalent operations (judged by the used Heuristic) are also outputted.

g

Output the graph graphically in gml-format after each step. The filename conventions are analogue to those from the GUI, except that in the case of an anonymous graph x will be equal to n, with n+1 the number of initial angular momenta, corresponding with a cubic graph of 2n nodes and 3n edges.

Mathematical and File Formats

Mathematical Standard Format

We will use the example above for describing all formats.

BRAKET Format

It is not needed to specify the intermediate angular momenta:
<((j1,j2),(j3,j4))j|((j1,j3),(j2,j4))j><EOF>
is also accepted. In addition the root label can be dropped too:
<((j1,j2),(j3,j4))|((j1,j3),(j2,j4))><EOF>
The intermediate angular momenta/root will get labels of the form t<number>, with <number> starting at 1. Note that in this case it is forbidden to use labels of this form.

YTS Format

<Header>
j1  j2  j5
j3  j4  j6
j5  j6  j7
j1  j3  j8
j2  j4  j9
j8  j9  j7<EOF>