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Cell[CellGroupData[{
Cell["Making Change and Finding Repfigits: Balancing a Knapsack", "Title"],
Cell["Daniel Lichtblau", "Author"],
Cell["\<\
100 Trade Center Dr
Champaign IL 61820
USA
danl@wolfram.com\
\>", "Address"],
Cell["\<\
Abstract. We will discuss knapsack problems that arise in certain \
computational number theory settings. A common theme is that the search space \
for the standard real relaxation is large; in a sense this translates to a \
poor choice of variables. Lattice reduction methods have been developed in \
the past few years to improve handling of such problems. We show explicitly \
how they may be applied to computation of Frobenius instances, Keith numbers \
(also called \"repfigits\"), and as a first step in computation of Frobenius \
numbers.
Key words. Frobenius instance solving, lattice reduction, integer linear \
programming, change-making problem, Frobenius numbers, Keith numbers, \
repfigits.\
\>", "Abstract"],
Cell[TextData[{
"A shorter, differently formatted version of this article appeared in ",
StyleBox["Proceedings of the Second International Congress on Mathematical \
Software (ICMS 2006)",
FontSlant->"Italic"],
", A. Iglesias and N. Takayama, eds. Lecture Notes in Computer Science ",
StyleBox["4151",
FontWeight->"Bold"],
":182\[Hyphen]193. Springer\[Hyphen]Verlag.2006."
}], "Text",
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". Introduction"
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Cell["\<\
Various problems in the realm of computational number theory have as key \
steps the solving of a linear equation or system over the integers, subject \
to some linear inequality constraints. The Frobenius instance problem (also \
known as the change\[Hyphen]making problem) is a well\[Hyphen]known example. \
The problem of finding what are called repfigits, to be described below, is \
another. These may be regarded as a class of knapsack problems wherein one is \
allowed to take only certain integer multiples of various items in forming a \
\"valid\" combination. As such these fall into the category of integer linear \
programming (ILP).\
\>", "Text"],
Cell[TextData[{
"Classical methods for solving such problems include branch\[Hyphen]and\
\[Hyphen]bound and cutting plane methods [",
CounterBox["Reference", "Dantzig"],
"] [",
CounterBox["Reference", "Schrijver"],
"]. These approaches alone are often inadequate for certain classes of \
problems due to a phenomenon roughly described as \"unbalanced bases\". A \
method for dealing with this deficiency was developed in [",
CounterBox["Reference", "Aardal Hurkens Lenstra"],
"] [",
CounterBox["Reference", "Aardal Lenstra"],
"]. In essence it involves working with a basis that is reduced (in the \
sense of [",
CounterBox["Reference", "Lenstra Lenstra Lovasz"],
"]) and ordered by size, in conjunction with standard branch\[Hyphen]and\
\[Hyphen]bound. We will describe and illustrate this for the types of problem \
mentioned above . We remark that the handling of Frobenius instances is by no \
means new, having been discussed in the aforementioned references. We also \
show how the method is applied in a new algorithm for finding Frobenius \
numbers, a task that is substantially harder than solving Frobenius instances."
}], "Text"],
Cell[TextData[{
"The algorithms described in this paper have been implemented in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" [",
CounterBox["Reference", "Wolfram"],
"]. Selected code is provided in the appendix."
}], "Text"]
}, Open ]],
Cell[CellGroupData[{
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". Frobenius Instances"
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Cell[TextData[{
"Suppose we have a set of positive integers ",
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" and a target ",
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", a positive integer. We seek nonnegative integer multipliers ",
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" such that ",
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". This is a standard problem in integer linear programming. A classical \
method [",
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"] for solving this would be to solve the relaxed problem wherein we enforce \
all inequality constraints but all variables to be real rather than integer \
valued. If in the solution we encounter a variable ",
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" with value ",
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" that is not an integer then we spawn two subproblems where we enforce \
respectively that ",
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" and ",
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". We continue this process of solving relaxed subproblems, splitting when a \
variable has a noninteger value. It can be shown that eventually either we \
exhaust all possibilities or we obtain an integer valued solution [",
CounterBox["Reference", "Schrijver"],
"]; in either case clearly the algorithm terminates."
}], "Text"],
Cell["\<\
A drawback to this approach is that the search space for the relaxed \
subproblems might appear to be \"large\", in the sense of having many points \
with not all coordinates integer valued. In particular it may be the case \
that a standard LP solver will find real valued solutions to the restricted \
subproblems without making rapid progress to an (entirely) integer valued \
solution, because it might be possible to subdivide the (real valued) \
solution polytope in such a way that integer points do not readily appear at \
corners.\
\>", "Text"],
Cell[TextData[{
"The method of [",
CounterBox["Reference", "Aardal Hurkens Lenstra"],
"], [",
CounterBox["Reference", "Aardal Lenstra"],
"] was developed as a way to improve on this situation. Roughly it proceeds \
as follows. First we find a description of a solution set to our equations \
that is a priori integer valued but possibly does not satisfy the required \
inequalities. We arrange that this solution set has \"good\" basis vectors, \
such that when we solve relaxed problems with these we more rapidly walk \
through our (real valued) solution polytope. More correctly, the polytope is \
likely to intersect fewer hyperplanes orthogonal to larger direction vectors \
and spaced by integral multiples of those vectors."
}], "Text"],
Cell[TextData[{
"With respect to the Frobenius instance problem it goes as follows. First we \
find a solution vector ",
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" over ",
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" for the integer null space (that is, ",
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" independent vectors ",
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"); for this we use a method based on the Hermite normal form [",
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"]. We use multiples of the basis vectors to find a \"small\" solution which \
we still call ",
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". The tactic utilized to do this is sometimes called the embedding method. \
It apparently has been independently discovered several times; variants \
appear in [",
CounterBox["Reference", "Aardal Hurkens Lenstra"],
"], [",
CounterBox["Reference", "Lichtblau 2003"],
"], [",
CounterBox["Reference", "Matthews"],
"], and [",
CounterBox["Reference", "Nguyen"],
"], with short code provided in the appendix of this paper. In starting with \
a solution over ",
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" rather than the nonnegatives ",
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" we are working with what is called an integer relaxation of the \
nonnegativity constraint. We will later enforce nonnegativity via the more \
common LP relaxations wherein integrality is not enforced."
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"We define variables ",
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" so that solution vectors are given by ",
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". For purposes of finding a valid solution to the problem at hand we need \
to impose two requirements. The first is that all components are nonnegative \
and the second is that all are integers. The first can be met by standard \
linear programming. For the second, as in the classical approach, we will use \
branching on subproblems. Specifically we find solutions to relaxed LP \
problems wherein we now work over nonnegative reals. We then branch on \
noninteger values in those solutions. For example, if a solution has, say, ",
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the new constraints ",
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", respectively (though note that if other variables also had noninteger \
solutions then we need not have chosen this particular one). As observed \
above, this branching process will terminate eventually, with either a valid \
solution or the information that no such solution exists."
}], "Text"],
Cell[TextData[{
"We explain again, in slightly different terms, why it is important to work \
with a small integer solution to the integer relaxation (that is, allowing \
negative values) and a lattice reduced basis for the null vectors. As our \
methodology is to take combinations of these null vectors, effectively they \
define directions in a solution polytope for the transformed problem. By \
working with a reduced basis we in effect change our coordinate system to one \
where the various search directions are roughly orthogonal. This helps us to \
avoid the possibility of taking many steps in similar directions in searching \
the polytope of nonnegative solutions for one that is integer valued. Thus we \
explore it far more efficiently. Moreover, in starting with a small solution \
we begin closer to the nonnegative orthant. Heuristically this seems to make \
the sought-for multipliers of the null vectors relatively small, and this is \
good for computational speed. This is discussed in section 2 of [",
CounterBox["Reference", "Aardal Hurkens Lenstra"],
"], with further explanation and illustrations found in [",
CounterBox["Reference", "Aardal Weismantel Wolsey"],
"]."
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"A further efficiency, from [",
CounterBox["Reference", "Aardal Lenstra"],
"], is to choose carefully the variable on which to branch. We order by \
increasing size the reduced lattice of null vectors. Branching will be done \
on the noninteger multiplier variable corresponding to the largest of these \
basis vectors. This has the effect of exploring the solution polytope in \
directions in which it is relatively thin, thus more quickly finding integer \
lattice points therein or exhausting the space. This refinement is important \
for handling pathological examples of the sort presented in [",
CounterBox["Reference", "Aardal Hurkens Lenstra"],
"] and [",
CounterBox["Reference", "Aardal Lenstra"],
"]."
}], "Text"],
Cell[TextData[{
"As we have a constraint satisfaction problem we are also free to impose any \
linear integer objective function of our choosing. Thus an optimization is to \
obtain extremal values for linear forms with integer coefficients and use \
these results as simple cutting planes [",
CounterBox["Reference", "Schrijver"],
"]. For example we can minimize or maximize the various coordinate values ",
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", selecting one either in some specific order or at random for each \
subproblem. This process amounts to finding the width of the polytope along \
the directions of our lattice basis vectors, and enforcing integrality of the \
optimized variable helps to further restrict the search space."
}], "Text"],
Cell[TextData[{
"As reported in [",
CounterBox["Reference", "Aardal Lenstra"],
"] this method is very effective in solving Frobenius instances. We \
illustrate with an example from [",
CounterBox["Reference", "Wagon Einstein Lichtblau Strzebonski"],
"]. "
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"We let the target be ",
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". In several seconds the instance solver returns the empty set. This has \
implications for bounding the Frobenius number of ",
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"; we will discuss that in a later section."
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". Keith Numbers"
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"Keith numbers, also known as repfigits, were introduced in 1987 by Michael \
Keith [",
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"] as a sort of computational novelty that relates a Fibonacci-like sequence \
to a linear equation involving its seed. They are defined as follows. Suppose \
we are given a number ",
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" of ",
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" digits (we work in base ",
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", but these can be defined with respect to arbitrary bases). We may form a \
sequence in Fibonacci style as follows. The first ",
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" elements are the digits themselves. The ",
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" element is the sum of the first ",
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" digits. Subsequent elements are the sums of the preceding ",
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" elements. Then ",
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" is called a Keith number provided it appears in this sequence. As an \
example, the sequence for ",
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" is ",
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" and so ",
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" is a Keith number. Keith originally referred to these as repfigits, for \
\"replicating Fibonacci digits\"."
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"Keith numbers tend to be quite rare (there are only ",
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" of them below ",
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"). Prior methods for finding them involved clever segmentation of an \
enumeration. While flawless (in the sense that they find all of them), these \
are limited in range due to algorithmic complexity and memory requirements. \
At the time the present work was begun the state of the art, from [",
CounterBox["Reference", "Keith"],
"], was that all such numbers up to ",
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" digits had been found but no larger ones were known. We will take this \
substantially further."
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"To begin we must find equations to describe these things. If the digits are \
",
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element up to the last by its successor, and replaces the last by the sum of \
the elements. For example, for ",
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"If we multiply this matrix by itself ",
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" times then the dot product of the bottom row with the digit sequence will \
give the ",
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fairly tight bounds on how many such multiples can possibly work for a given \
number of digits ",
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". We will use each possibility to form a homogeneous linear diophantine \
equation. In the actual code we take advantage of the structure of the matrix \
to avoid forming explicit matrix products."
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"We demonstrate with a short example. We start by obtaining the set of \
candidate equation vectors for ",
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" digit examples. One of them is ",
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"As in the last section, the first step in the process of [",
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"] and [",
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"] is to find a full set of integer solutions to such a system. Since these \
are homogeneous equations we require only the integer null space. This can be \
obtained readily from the Hermite normal form for the matrix comprised of the \
vector for the homogeneous equation, augmented by an identity matrix. Again \
we want to work with vectors that are small and close to orthogonal so we \
apply lattice reduction to get a \"good\" set of vectors spanning the same \
solution set. We obtain ",
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"Notice that for any solution vector, its negative is also a solution \
vector. Looking at the first vector in our solution basis we thus see that ",
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" is a Keith number of five digits. That was too easy; there is no guarantee \
we will have a solution vector with all components in the desired range. We \
look at a slightly larger example to see this. For six digits one candidate \
equation\[Hyphen]defining vector is ",
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". So we have a generating set of small null vectors none of which have \
entirely nonnegative or entirely nonpositive values (with the first being \
nonzero, in order that they give a legitimate six digit number). We now need \
a way to recombine these so that the first component is positive and the rest \
are nonnegative."
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"Again we have something we can tackle with standard branch-and-bound \
iterations [",
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" be our null space basis (here ",
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" is one less than the number of digits). We seek an integer vector of the \
form ",
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",
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" which will ultimately be required to take on integer values. In order to \
effect this we will solve relaxed linear programming problems with \
appropriate inequality constraints. As noted earlier, these are simply \
constraint satisfaction problems, so we can use arbitrary linear objective \
functions as a means of obtaining integer cuts cheaply. When some variables \
do not take on integer values in the solution we choose one such on which to \
branch and spawn a pair of subproblems. Our choice again is from [",
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corresponding basis vector is largest."
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Cell["\<\
To summarize, we first find the appropriate sets of integer equations. For \
each we find spanning sets of solutions that do not in general satisfy the \
digit inequality constraints. We lattice reduce these. We use ILP methods to \
find all possible solutions subject to the usual inequality constraints on \
digits.\
\>", "Text"],
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Code that implements this is found in the appendix. We used this to find all \
Keith numbers up through 29 digits. We show all repfigits between 20 and 29 \
digits below.\
\>", "Text"],
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Cell[TextData[{
"We remark that lattice methods alone can find sporadic large Keith numbers. \
One approach, from [",
CounterBox["Reference", "Schnorr Euchner"],
"], improves the chances of getting a valid result from the lattice \
reduction step. The idea is to augment each null vector with a zero, and \
augment the lattice with a row consisting of some nonzero value (typically \
one) in the new column of zeros, and ",
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" everywhere else. Thus if there is a valid solution then this augmented \
lattice contains the vector consisting of that nonzero value (or its \
negative) and the remaining entries in the range ",
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". As this would be a fairly \"small\" vector, one can hope that it will \
appear in the reduced basis (this is essentially the idea used by Schnorr and \
Euchner, in a binary setting, to raise the density at which one can typically \
solve subset sum problems). In practice we get a few Keith numbers this way \
as well as several more near misses."
}], "Text"],
Cell["\<\
It might be effective to combine this different lattice formulation with the \
branch\[Hyphen]and\[Hyphen]bound regimen described above. This is not \
entirely trivial as that required that the vectors span a solution space for \
a homogeneous equation, whereas the vectors in this modified lattice need not \
satisfy that equation.\
\>", "Text"],
Cell[TextData[{
"Observe that the Keith numbers beginning at 24 digits but smaller than ",
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" digits all have leading digit in the set ",
Cell[BoxData[
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trend returns after ",
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" digits. Also all known Keith numbers from 25 digits onward have a final \
digit that is either even or 5, and hence they cannot be prime. Again, one \
might wonder whether this trend continues, and, if so, whether there is an \
interesting reason behind it."
}], "Text"],
Cell[TextData[{
"We will say a bit about the practical complexity of the method we have \
described for solving ILPs. While in principle branching can have very bad \
performance, in practice we find that the complexity scales reasonably well \
with problem size. Although we cannot claim that the methods described in \
this paper are polynomial time even in fixed dimension, in practice they do \
seem to scale that way. The behavior with respect to dimension is of course \
not so nice. For Keith number computations we find that, on average, the time \
spent for handling ",
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" digits. Considering that each additional digit multiplies the search space \
by a factor of ",
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" this is still not so bad. To give an indication of computational speed, \
the implementation in the appendix was able to handle ",
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" digits in around three days on a 3.0 GHz machine. We emphasize that this \
is but a crude measure both of complexity and actual performance, as various \
sorts of optimizations could have a significant impact on each. For example, \
preliminary experiments with the software in [",
CounterBox["Reference", "COIN"],
"] indicate room for improvement in the handling of the ILP solving after \
the lattice reduction phase."
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Cell[CellGroupData[{
Cell[TextData[{
CounterBox["Section"],
". Frobenius Numbers"
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Cell[TextData[{
"We are given a set ",
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" of positive integers with ",
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". For later purposes we assume that the set is in ascending order. It can \
be shown that there are at most finitely many numbers not representable as a \
nonnegative integer combination of elements in ",
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". The largest such nonrepresentable is called the Frobenius number of the \
set. The \"Frobenius number problem\" is to find it. Generally speaking this \
tends to be a different and usually harder problem than the Frobenius \
instance solving discussed earlier. We will give a much abbreviated \
discussion here in order to indicate how the sort of knapsack solving under \
discussion plays a role in computation of these numbers. References may be \
found in [",
CounterBox["Reference", "Beihoffer Nijinhuis Hendry Wagon"],
"] and [",
CounterBox["Reference", "Wagon Einstein Lichtblau Strzebonski"],
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was popularized by Frobenius early in the twentieth century). Moreover in the \
1980's some very good methods appeared for the case ",
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generalization that subsumes both lattice diagrams in earlier literature and \
a circulant graph description from [",
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"]. Similar ideas, expressed in the terminology of Minimal Distance \
Diagrams, appear in [",
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origin, in a suitably weighted ",
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"We start with the lattice of integer combinations of ",
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that there are ",
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now define the \"weight\" of a vector ",
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all nonnegative entries. From those we choose one of minimal weight. In case \
of a tie we choose the one that is lexicographically last. This uniquely \
defines the set of residues that we take to comprise the fundamental domain."
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"This domain has several interesting properties. (1) It is a staircase: if \
it contains a lattice element then it contains all nonnegative vectors with \
any coordinate strictly smaller. (2) It tiles ",
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that was utilized in various shortest\[Hyphen]path graph methods. Old and new \
approaches using such methods are discussed at length in [",
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"From the staircase property the fundamental domain has turning points we \
refer to as elbows. It has extremal points called corners. Specifically, a \
corner is a point ",
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\"axial\" elbow. It indicates how far one can go along a given axis and still \
remain inside the domain. There are two other definitions that play a role in \
the algorithm. We will not descibe them too carefully but, roughly, there are \
as follows."
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(1) Protoelbows. These have both positive and negative coordinates and \
correspond to certain \"minimal\" equivalences (that is, reducing relations) \
in the lattice.\
\>", "Text"],
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(2) Preelbows. These are the positive parts of the protoelbows. Elbows are \
minimal elements in the partially ordered (ascending by inclusion) set of \
preelbows.\
\>", "Text"],
Cell[TextData[{
"With respect to these domains the Frobenius number corresponds to the \
farthest corner from the origin where distance is an ",
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Recall that each element in the domain corresponds to a residue modulo ",
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residue class but of smaller weight cannot be attained using nonnegative \
combinations of elements of ",
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combinations. We conclude that the largest nonattainable value for the \
residue class of the element ",
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"With this brief background we now say a bit about how knapsack solving can \
play a role in the computation of Frobenius numbers. First, it turns out that \
axial elbows are defined by an integer programming problem (see [",
CounterBox["Reference", "Wagon Einstein Lichtblau Strzebonski"],
"]) which, in complexity if not details of definition, is similar to the \
Frobenius instance problem. From the axial elbows we immediately get good \
lower and upper bounds on the Frobenius number (each elbow less one is in the \
domain, and the farthest corner is bounded by the vector with all components \
given by corresponding axial elbows less one). More importantly for our \
purposes is that they give a search space from which to find all elbows. They \
are particular integer points in a polyhedron that satisfy certain inequality \
conditions. Full details, including a method for finding them, are provided \
in [",
CounterBox["Reference", "Wagon Einstein Lichtblau Strzebonski"],
"]. From the elbows one can find all corners and in particular the one that \
gives the Frobenius number of the set."
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"Another tactic presented in [",
CounterBox["Reference", "Wagon Einstein Lichtblau Strzebonski"],
"] makes direct use of Frobenius instance solving to find axial elbows. One \
uses a bisection approach, working down from an a priori bound on the axial \
elbow values. The goal is to find the smallest value for which a certain set \
of Frobenius instances have no solution."
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Cell[TextData[{
"Still another point of overlap between Frobenius instances and numbers is \
the obvious fact that whenever a Frobenius instance solver returns an empty \
solution we automatically have a lower bound on the Frobenius number. One can \
test random values that are, say, an order of magnitude below the heuristic \
approximation for the Frobenius number presented in [",
CounterBox["Reference", "Beihoffer Nijinhuis Hendry Wagon"],
"]. If any such test gives no solution we thereby establish a lower bound \
that is often better than a priori bounds to be found in the literature."
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"There is also a heuristic method in [",
CounterBox["Reference", "Wagon Einstein Lichtblau Strzebonski"],
"] for more efficiently \"guessing\" the likely Frobenius number from a \
restricted set of elbows. It gives an a priori upper bound, and a single \
Frobenius instance invocation can then verify whether it is in fact the \
actual value. In random examples this appears always to be the case."
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Cell[TextData[{
"We sketched above how efficient ILP knapsack solving, of the sort used for \
Frobenius instances, also may be applied to the (generally much harder) \
problem of finding Frobenius numbers. From the fundamental domain pictures \
one realizes a possible alternative approach. Working an axis at a time, a \
branch\[Hyphen]and\[Hyphen]bound strategy might be directly applied to get to \
extremal vertices (corners) in the domain. Thus we could have a bilevel \
branching algorithm, with the outer level iterating over these extrema, and \
the inner one using relaxed LPs to solve the ILPs needed to move to new \
vertices. This might become an alternative to the algorithm described in [",
CounterBox["Reference", "Wagon Einstein Lichtblau Strzebonski"],
"]."
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Cell[TextData[{
"We remark that there is a connection between another knapsack solving \
technique and the problem of computing Frobenius numbers. It is well known \
that integer programming e.g. for knapsack problems can be done with toric Gr\
\[ODoubleDot]bner bases [",
CounterBox["Reference", "Conti Traverso"],
"]. What is not so obvious is that they may also be used to deduce the \
stairway structure of the fundamental domain. An algorithm for this purpose \
is presented in [",
CounterBox["Reference", "Wagon Einstein Lichtblau Strzebonski"],
"]. The basic idea is to formulate a term ordering so that the staircase \
structure of the fundamental domain is captured by the staircase of the Gr\
\[ODoubleDot]bner basis lead monomials. This has the added virtue of finding \
all elbows at once, so no elaborate method is needed to search a bounding box \
defined by axial elbows. An implementation by the author has handled \
Frobenius number problems involving as many as ",
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" numbers of ",
Cell[BoxData[
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" digits. While it is not competitive with the main approach in [",
CounterBox["Reference", "Wagon Einstein Lichtblau Strzebonski"],
"] (which has handled sets of up to ",
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" numbers), Frobenius number problems of this size are apparently larger \
than what can be handled by other methods from the published literature."
}], "Text"],
Cell[TextData[{
"Another link between Frobenius numbers and methods from ideal theory \
appears implicitly in [",
CounterBox["Reference", "Gomez Gutierrez Ibeas"],
"] as well as other literature concerning what are called \
\"multiple\[Hyphen]loop networks\". There, as in [",
CounterBox["Reference", "Beihoffer Nijinhuis Hendry Wagon"],
"], one works with a circulant directed graph based on residues modulo a \
positive integer. Now, however, the modulus is the largest rather than \
smallest element of the given set. While the authors did not explicitly \
consider the connection to Frobenius numbers, some of the ideas are quite \
similar. In particular the maximum diameter of this graph, which is similar \
to the Frobenius number (though using an unweighted metric), plays an \
important role in their work. They discuss several aspects of the related \
domain (actually family of domains, as they do not impose uniqueness \
conditions) in terms of monomial ideals. They define a generalized ell shape \
and prove their domains are always of such a shape; this corresponds to the \
uniqueness of the interior elbow as shown in [",
CounterBox["Reference", "Wagon Einstein Lichtblau Strzebonski"],
"]. It would be interesting to understand better how their ideas, in \
particular regarding use of monomial ideals, relate to the toric ideal \
construction of the fundamental domain given in [",
CounterBox["Reference", "Wagon Einstein Lichtblau Strzebonski"],
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Cell["\<\
We have investigated several examples of integer linear programs with the \
common feature that a straightforward branch\[Hyphen]and\[Hyphen]bound \
approach, working with real relaxations, will tend to bog down in searching a \
large polytope. We utilize a method based on solving of an integer relaxation \
to the set of equality constraints. We reformulate the problem as one of \
adding combinations of null vectors to a specific solution. The vectors in \
question tend to be well suited to the problem at hand because we use lattice \
reduction to make them close to orthogonal. We then enforce inequality \
constraints via branch\[Hyphen]and\[Hyphen]bound on real relaxations of the \
new problem. We use a branching choice that tends to make the polytope thin \
in the search direction and thus helps to exhaust it efficiently.\
\>", "Text"],
Cell[TextData[{
"We reviewed this approach as it was applied to the change\[Hyphen]making \
problem [",
CounterBox["Reference", "Aardal Hurkens Lenstra"],
"] [",
CounterBox["Reference", "Aardal Lenstra"],
"]. We then used it to find all Keith numbers through 29 digits; previous \
methods had gone only through 19 digits. This brought us to a range where we \
could observe curious patterns in leading and trailing digits, something not \
present in the smaller Keith numbers. We also gave a brief idea of how this \
method is used in a new algorithm to compute Frobenius numbers."
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Cell[TextData[{
"A further direction would be to incorporate effective cutting planes. The \
code in the appendix only attempts a very naive sort of cut (by using random \
coordinate variables as objective function). Preliminary experiments with an \
external library [",
CounterBox["Reference", "COIN"],
"] indicate that more serious cutting plane efforts can give substantial \
speed improvement; we have seen ILP examples that improve by an order of \
magnitude. We emphasize that this still requires preprocessing with lattice \
reduction in the manner described in this paper."
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Cell[CellGroupData[{
Cell[" Acknowledgements", "Subsubsection",
CellDingbat->None],
Cell[TextData[{
"I thank Stan Wagon for providing the diagrams of fundamental domains and \
for his careful reading and many suggestions that improved the exposition. I \
thank Victor Moll for inviting me to attend the 2005 Clifford Lectures \
conference at Tulane, where I presented an earlier version of this work. I \
thank the two anonymous referees for their several suggestions which improved \
readability, and the second referee moreover for pointing out the relevance \
of [",
CounterBox["Reference", "Gomez Gutierrez Ibeas"],
"] and related literature on multiple\[Hyphen]loop networks."
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" Implementation of Selected Algorithms"
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We solve a system of integer equations over integers and extract a small \
solution by using combinations of null vectors to decrease the size of a \
specific solution. This is used as a first step in solving Frobenius \
instances, for example.\
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The code below will find the set of Keith number linear equations for a given \
number of digits. Integer solutions to any of these equations, subject to the \
constraints that the first variable be positive and all variables lie between \
0 and 9, will give Keith numbers.\
\>", "Text"],
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Here we give a fairly straightforward implementation of the Keith number \
solver. The input is a set of integer vectors spanning the solution space for \
a particular Keith number homogeneous linear equation. The program will find \
all possible combinations that have all components between 0 and 9 and the \
first one nonzero (so they correspond to digits with the leading one \
positive), or else terminate with an empty solution set.\
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Not surprisingly there are various ways to improve on this sort of solving \
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The next snippets of code will generate all possible Keith number equations \
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}], "Section"],
Cell[TextData[{
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\>", "Reference",
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}], "Reference",
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Cell[TextData[{
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". Electronic manuscript, 1998.\nAvailable electronically at:\n\
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CellMargins -> {{9, Inherited}, {Inherited, Inherited}},
StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize ->
9, FontSlant -> "Oblique"],
Cell[
StyleData["CellLabel", "Presentation"], FontSize -> 14],
Cell[
StyleData["CellLabel", "Printout"],
CellMargins -> {{0, Inherited}, {Inherited, Inherited}},
FontSize -> 8]}, Closed]]}, Open]],
Cell[
CellGroupData[{
Cell["Unique Styles", "Section"],
Cell[
CellGroupData[{
Cell[
StyleData["Author"], CellMargins -> {{-20, Inherited}, {2, 20}},
CellGroupingRules -> {"TitleGrouping", 20}, PageBreakBelow ->
False, TextAlignment -> Center,
CounterAssignments -> {{"Section", 0}, {"Equation", 0}, {
"Figure", 0}}, FontSize -> 14, FontWeight -> "Bold"],
Cell[
StyleData["Author", "Presentation"],
CellMargins -> {{15, 30}, {4, 30}}, FontSize -> 21],
Cell[
StyleData["Author", "Printout"],
CellMargins -> {{-20, Inherited}, {2, 14}}, FontSize -> 12]},
Open]],
Cell[
CellGroupData[{
Cell[
StyleData["Address"], CellMargins -> {{-20, Inherited}, {2, 2}},
CellGroupingRules -> {"TitleGrouping", 30}, PageBreakBelow ->
False, TextAlignment -> Center, LineSpacing -> {1, 1},
CounterAssignments -> {{"Section", 0}, {"Equation", 0}, {
"Figure", 0}}, FontSize -> 12, FontSlant -> "Italic"],
Cell[
StyleData["Address", "Presentation"],
CellMargins -> {{15, 30}, {40, 2}}, FontSize -> 18],
Cell[
StyleData["Address", "Printout"],
CellMargins -> {{-20, Inherited}, {2, 2}}, FontSize -> 10]},
Open]],
Cell[
CellGroupData[{
Cell[
StyleData["Abstract"],
CellMargins -> {{45, 75}, {Inherited, 30}},
LineSpacing -> {1, 0}],
Cell[
StyleData["Abstract", "Presentation"],
CellMargins -> {{65, 30}, {8, 25}}, FontSize -> 18],
Cell[
StyleData["Abstract", "Printout"],
CellMargins -> {{36, 67}, {Inherited, 20}}, Hyphenation -> True,
FontSize -> 9.5]}, Open]],
Cell[
CellGroupData[{
Cell[
StyleData["Reference"], CellMargins -> {{18, 40}, {2, 2}},
CellFrameLabels -> {{
Cell[
TextData[{"[",
CounterBox["Reference"], "] "}], "Reference", CellBaseline ->
Baseline], None}, {None, None}}, TextJustification -> 1,
Hyphenation -> True, LineSpacing -> {1, 1}, CounterIncrements ->
"Reference", CounterStyleMenuListing -> Automatic],
Cell[
StyleData["Reference", "Presentation"],
CellMargins -> {{25, 40}, {2, 2}}, FontSize -> 18],
Cell[
StyleData["Reference", "Printout"],
CellMargins -> {{14, 16}, {Inherited, 0}}, Hyphenation -> True,
FontSize -> 9]}, Open]]}, Open]],
Cell[
CellGroupData[{
Cell["Hyperlink Styles", "Section"],
Cell[
"The cells below define styles useful for making hypertext \
ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \
or between Notebooks.", "Text"],
Cell[
CellGroupData[{
Cell[
StyleData["Hyperlink"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic, FontColor ->
RGBColor[0.269993, 0.308507, 0.6],
ButtonBoxOptions -> {
Active -> True, Appearance -> {Automatic, None},
ButtonFunction :> (FrontEndExecute[{
FrontEnd`NotebookLocate[#2]}]& ), ButtonNote ->
ButtonData}],
Cell[
StyleData["Hyperlink", "Presentation"], FontSize -> 16],
Cell[
StyleData["Hyperlink", "Condensed"], FontSize -> 11],
Cell[
StyleData["Hyperlink", "SlideShow"]],
Cell[
StyleData["Hyperlink", "Printout"], FontSize -> 10,
FontVariations -> {"Underline" -> False}, FontColor ->
GrayLevel[0]]}, Closed]],
Cell[
"The following styles are for linking automatically to the on-line \
help system.", "Text"],
Cell[
CellGroupData[{
Cell[
StyleData["MainBookLink"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic, FontColor ->
RGBColor[0.269993, 0.308507, 0.6],
ButtonBoxOptions -> {
Active -> True, Appearance -> {Automatic, None},
ButtonFunction :> (FrontEndExecute[{
FrontEnd`HelpBrowserLookup["MainBook", #]}]& )}],
Cell[
StyleData["MainBookLink", "Presentation"], FontSize -> 16],
Cell[
StyleData["MainBookLink", "Condensed"], FontSize -> 11],
Cell[
StyleData["MainBookLink", "SlideShow"]],
Cell[
StyleData["MainBookLink", "Printout"], FontSize -> 10,
FontVariations -> {"Underline" -> False}, FontColor ->
GrayLevel[0]]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["AddOnsLink"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic, FontFamily -> "Courier",
FontColor -> RGBColor[0.269993, 0.308507, 0.6],
ButtonBoxOptions -> {
Active -> True, Appearance -> {Automatic, None},
ButtonFunction :> (FrontEndExecute[{
FrontEnd`HelpBrowserLookup["AddOns", #]}]& )}],
Cell[
StyleData["AddOnsLink", "Presentation"], FontSize -> 16],
Cell[
StyleData["AddOnsLink", "Condensed"], FontSize -> 11],
Cell[
StyleData["AddOnsLink", "SlideShow"]],
Cell[
StyleData["AddOnsLink", "Printout"], FontSize -> 10,
FontVariations -> {"Underline" -> False}, FontColor ->
GrayLevel[0]]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["RefGuideLink"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic, FontFamily -> "Courier",
FontColor -> RGBColor[0.269993, 0.308507, 0.6],
ButtonBoxOptions -> {
Active -> True, Appearance -> {Automatic, None},
ButtonFunction :> (FrontEndExecute[{
FrontEnd`HelpBrowserLookup["RefGuide", #]}]& )}],
Cell[
StyleData["RefGuideLink", "Presentation"], FontSize -> 16],
Cell[
StyleData["RefGuideLink", "Condensed"], FontSize -> 11],
Cell[
StyleData["RefGuideLink", "SlideShow"]],
Cell[
StyleData["RefGuideLink", "Printout"], FontSize -> 10,
FontVariations -> {"Underline" -> False}, FontColor ->
GrayLevel[0]]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["RefGuideLinkText"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic, FontColor ->
RGBColor[0.269993, 0.308507, 0.6],
ButtonBoxOptions -> {
Active -> True, Appearance -> {Automatic, None},
ButtonFunction :> (FrontEndExecute[{
FrontEnd`HelpBrowserLookup["RefGuide", #]}]& )}],
Cell[
StyleData["RefGuideLinkText", "Presentation"], FontSize -> 16],
Cell[
StyleData["RefGuideLinkText", "Condensed"], FontSize -> 11],
Cell[
StyleData["RefGuideLinkText", "SlideShow"]],
Cell[
StyleData["RefGuideLinkText", "Printout"], FontSize -> 10,
FontVariations -> {"Underline" -> False}, FontColor ->
GrayLevel[0]]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["GettingStartedLink"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic, FontColor ->
RGBColor[0.269993, 0.308507, 0.6],
ButtonBoxOptions -> {
Active -> True, Appearance -> {Automatic, None},
ButtonFunction :> (FrontEndExecute[{
FrontEnd`HelpBrowserLookup["GettingStarted", #]}]& )}],
Cell[
StyleData["GettingStartedLink", "Presentation"], FontSize -> 16],
Cell[
StyleData["GettingStartedLink", "Condensed"], FontSize -> 11],
Cell[
StyleData["GettingStartedLink", "SlideShow"]],
Cell[
StyleData["GettingStartedLink", "Printout"], FontSize -> 10,
FontVariations -> {"Underline" -> False}, FontColor ->
GrayLevel[0]]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["DemosLink"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic, FontColor ->
RGBColor[0.269993, 0.308507, 0.6],
ButtonBoxOptions -> {
Active -> True, Appearance -> {Automatic, None},
ButtonFunction :> (FrontEndExecute[{
FrontEnd`HelpBrowserLookup["Demos", #]}]& )}],
Cell[
StyleData["DemosLink", "SlideShow"]],
Cell[
StyleData["DemosLink", "Printout"],
FontVariations -> {"Underline" -> False}, FontColor ->
GrayLevel[0]]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["TourLink"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic, FontColor ->
RGBColor[0.269993, 0.308507, 0.6],
ButtonBoxOptions -> {
Active -> True, Appearance -> {Automatic, None},
ButtonFunction :> (FrontEndExecute[{
FrontEnd`HelpBrowserLookup["Tour", #]}]& )}],
Cell[
StyleData["TourLink", "SlideShow"]],
Cell[
StyleData["TourLink", "Printout"],
FontVariations -> {"Underline" -> False}, FontColor ->
GrayLevel[0]]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["OtherInformationLink"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic, FontColor ->
RGBColor[0.269993, 0.308507, 0.6],
ButtonBoxOptions -> {
Active -> True, Appearance -> {Automatic, None},
ButtonFunction :> (FrontEndExecute[{
FrontEnd`HelpBrowserLookup["OtherInformation", #]}]& )}],
Cell[
StyleData["OtherInformationLink", "Presentation"], FontSize ->
16],
Cell[
StyleData["OtherInformationLink", "Condensed"], FontSize -> 11],
Cell[
StyleData["OtherInformationLink", "SlideShow"]],
Cell[
StyleData["OtherInformationLink", "Printout"], FontSize -> 10,
FontVariations -> {"Underline" -> False}, FontColor ->
GrayLevel[0]]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["MasterIndexLink"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic, FontColor ->
RGBColor[0.269993, 0.308507, 0.6],
ButtonBoxOptions -> {
Active -> True, Appearance -> {Automatic, None},
ButtonFunction :> (FrontEndExecute[{
FrontEnd`HelpBrowserLookup["MasterIndex", #]}]& )}],
Cell[
StyleData["MasterIndexLink", "SlideShow"]],
Cell[
StyleData["MasterIndexLink", "Printout"],
FontVariations -> {"Underline" -> False}, FontColor ->
GrayLevel[0]]}, Closed]]}, Closed]],
Cell[
CellGroupData[{
Cell["Palette Styles", "Section"],
Cell[
"The cells below define styles that define standard ButtonFunctions, \
for use in palette buttons.", "Text"],
Cell[
StyleData["Paste"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic,
ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{
FrontEnd`NotebookApply[
FrontEnd`InputNotebook[], #, After]}]& )}],
Cell[
StyleData["Evaluate"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic,
ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{
FrontEnd`NotebookApply[
FrontEnd`InputNotebook[], #, All],
FrontEnd`SelectionEvaluate[
FrontEnd`InputNotebook[], All]}]& )}],
Cell[
StyleData["EvaluateCell"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic,
ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{
FrontEnd`NotebookApply[
FrontEnd`InputNotebook[], #, All],
FrontEnd`SelectionMove[
FrontEnd`InputNotebook[], All, Cell, 1],
FrontEnd`SelectionEvaluateCreateCell[
FrontEnd`InputNotebook[], All]}]& )}],
Cell[
StyleData["CopyEvaluate"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic,
ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{
FrontEnd`SelectionCreateCell[
FrontEnd`InputNotebook[], All],
FrontEnd`NotebookApply[
FrontEnd`InputNotebook[], #, All],
FrontEnd`SelectionEvaluate[
FrontEnd`InputNotebook[], All]}]& )}],
Cell[
StyleData["CopyEvaluateCell"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic,
ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{
FrontEnd`SelectionCreateCell[
FrontEnd`InputNotebook[], All],
FrontEnd`NotebookApply[
FrontEnd`InputNotebook[], #, All],
FrontEnd`SelectionEvaluateCreateCell[
FrontEnd`InputNotebook[], All]}]& )}]}, Closed]],
Cell[
CellGroupData[{
Cell["Slide Show Styles", "Section"],
Cell[
CellGroupData[{
Cell[
StyleData["SlideShowNavigationBar"], Editable -> False,
CellFrame -> True, CellMargins -> {{0, 0}, {3, 3}},
CellElementSpacings -> {"CellMinHeight" -> 0.8125},
CellGroupingRules -> {"SectionGrouping", 30}, CellFrameMargins ->
False, CellFrameColor -> GrayLevel[1], CellFrameLabelMargins ->
False, TextAlignment -> Center, CounterIncrements ->
"SlideShowNavigationBar", StyleMenuListing -> None, FontSize ->
10, Magnification -> 1, Background -> GrayLevel[0.8],
GridBoxOptions -> {
BaselinePosition -> Center,
GridBoxAlignment -> {
"Columns" -> {
Center, Center, Center, Center, Center, Center, Right, {
Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}},
"RowsIndexed" -> {}},
GridBoxItemSize -> {
"Columns" -> {3.5, 3.5, 3.5, 3.5, 13, 5, {4}},
"ColumnsIndexed" -> {}, "Rows" -> {{1.}},
"RowsIndexed" -> {}}, GridBoxSpacings -> {"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.]},
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {},
"Rows" -> {
Offset[0.2], {
Offset[0.]},
Offset[0.2]}, "RowsIndexed" -> {}}}],
Cell[
StyleData["SlideShowNavigationBar", "Presentation"]],
Cell[
StyleData["SlideShowNavigationBar", "SlideShow"], Deletable ->
False, ShowCellBracket -> False,
CellMargins -> {{-1, -1}, {-1, -1}}, PageBreakAbove -> True,
CellFrameMargins -> {{1, 1}, {0, 0}}],
Cell[
StyleData["SlideShowNavigationBar", "Printout"],
CellMargins -> {{18, 4}, {4, 4}}, LineSpacing -> {1, 3},
FontSize -> 10]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["SlideShowSection"], CellFrame -> {{0, 0}, {0, 0.5}},
CellMargins -> {{0, 0}, {10, 0}},
CellGroupingRules -> {"SectionGrouping", 40}, PageBreakBelow ->
False, CellFrameMargins -> {{12, 4}, {6, 12}},
InputAutoReplacements -> {"TeX" -> StyleBox[
RowBox[{"T",
AdjustmentBox[
"E", BoxMargins -> {{-0.075, -0.085}, {0, 0}},
BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[
RowBox[{"L",
StyleBox[
AdjustmentBox[
"A", BoxMargins -> {{-0.36, -0.1}, {0, 0}},
BoxBaselineShift -> -0.2], FontSize -> Smaller], "T",
AdjustmentBox[
"E", BoxMargins -> {{-0.075, -0.085}, {0, 0}},
BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica",
"Mma" -> "Mathematica", "MMA" -> "Mathematica",
"gridMathematica" -> FormBox[
RowBox[{"grid",
AdjustmentBox[
StyleBox["Mathematica", FontSlant -> "Italic"],
BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm],
"webMathematica" -> FormBox[
RowBox[{"web",
AdjustmentBox[
StyleBox["Mathematica", FontSlant -> "Italic"],
BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm],
Inherited}, CounterIncrements -> "Section",
CounterAssignments -> {{"Subsection", 0}, {"Subsubsection", 0}},
StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize ->
18, FontWeight -> "Plain", FontColor -> GrayLevel[1], Background ->
RGBColor[0.408011, 0.440726, 0.8]],
Cell[
StyleData["SlideShowSection", "Presentation"],
CellFrameMargins -> {{18, 10}, {10, 18}}, FontSize -> 27],
Cell[
StyleData["SlideShowSection", "SlideShow"], ShowCellBracket ->
False, PageBreakAbove -> True],
Cell[
StyleData["SlideShowSection", "Printout"],
CellMargins -> {{18, 30}, {0, 30}}, CellFrameMargins -> 5,
FontSize -> 14]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["SlideHyperlink"], StyleMenuListing -> None,
ButtonStyleMenuListing -> Automatic, FontSize -> 26, FontColor ->
GrayLevel[0.400015],
ButtonBoxOptions -> {
Active -> True, ButtonFunction :> (FrontEndExecute[{
FrontEnd`NotebookLocate[#2]}]& ), ButtonMargins -> 0.5,
ButtonMinHeight -> 0.85, ButtonNote -> None}],
Cell[
StyleData["SlideHyperlink", "Presentation"],
CellMargins -> {{10, 10}, {10, 12}}, FontSize -> 36],
Cell[
StyleData["SlideHyperlink", "SlideShow"], FontSize -> 26],
Cell[
StyleData["SlideHyperlink", "Printout"], FontSize -> 10,
FontVariations -> {"Underline" -> False}, FontColor ->
GrayLevel[0]]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["SlideTOCLink"],
CellMargins -> {{24, Inherited}, {Inherited, Inherited}},
StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic,
FontFamily -> "Helvetica",
ButtonBoxOptions -> {
Active -> True, ButtonFunction :> (FrontEndExecute[{
FrontEnd`NotebookLocate[#2]}]& ), ButtonMargins -> 1.5,
ButtonNote -> ButtonData}],
Cell[
StyleData["SlideTOCLink", "Presentation"],
CellMargins -> {{35, 10}, {15, 12}}, FontSize -> 18],
Cell[
StyleData["SlideTOCLink", "SlideShow"], FontSize -> 12],
Cell[
StyleData["SlideTOCLink", "Printout"],
FontVariations -> {"Underline" -> False}, FontColor ->
GrayLevel[0]]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["SlideTOC"], CellDingbat -> "\[Bullet]",
CellMargins -> {{18, Inherited}, {Inherited, Inherited}},
StyleMenuListing -> None, FontFamily -> "Helvetica"],
Cell[
StyleData["SlideTOC", "Presentation"],
CellMargins -> {{25, 10}, {10, 10}}, FontSize -> 18],
Cell[
StyleData["SlideTOC", "SlideShow"], FontSize -> 14],
Cell[
StyleData["SlideTOC", "Printout"], FontSize -> 10, FontColor ->
GrayLevel[0]]}, Closed]]}, Closed]],
Cell[
CellGroupData[{
Cell["Styles for Automatic Numbering", "Section"],
Cell[
"The following styles are useful for numbered equations, figures, \
etc. They automatically give the cell a FrameLabel containing a reference to \
a particular counter, and also increment that counter.", "Text"],
Cell[
CellGroupData[{
Cell[
StyleData["NumberedEquation"],
CellMargins -> {{55, 10}, {0, 10}}, CellFrameLabels -> {{None,
Cell[
TextData[{"(",
CounterBox["NumberedEquation"], ")"}]]}, {None, None}},
DefaultFormatType -> DefaultInputFormatType,
"TwoByteSyntaxCharacterAutoReplacement" -> True,
HyphenationOptions -> {
"HyphenationCharacter" -> "\[Continuation]"}, CounterIncrements ->
"NumberedEquation", FormatTypeAutoConvert -> False],
Cell[
StyleData["NumberedEquation", "Presentation"],
CellMargins -> {{80, 10}, {0, 20}}, FontSize -> 18],
Cell[
StyleData["NumberedEquation", "Printout"],
CellMargins -> {{55, 55}, {0, 10}}, FontSize -> 10]}, Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["NumberedFigure"], CellMargins -> {{55, 145}, {2, 10}},
CellHorizontalScrolling -> True,
CellFrameLabels -> {{None, None}, {
Cell[
TextData[{"Figure ",
CounterBox["NumberedFigure"]}], FontWeight -> "Bold"],
None}}, CounterIncrements -> "NumberedFigure",
FormatTypeAutoConvert -> False],
Cell[
StyleData["NumberedFigure", "Presentation"],
CellMargins -> {{80, 10}, {0, 20}}, FontSize -> 18],
Cell[
StyleData["NumberedFigure", "Printout"], FontSize -> 10]},
Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["NumberedTable"], CellMargins -> {{55, 145}, {2, 10}},
CellFrameLabels -> {{None, None}, {
Cell[
TextData[{"Table ",
CounterBox["NumberedTable"]}], FontWeight -> "Bold"],
None}}, TextAlignment -> Center, CounterIncrements ->
"NumberedTable", FormatTypeAutoConvert -> False],
Cell[
StyleData["NumberedTable", "Presentation"],
CellMargins -> {{80, 10}, {0, 20}}, FontSize -> 18],
Cell[
StyleData["NumberedTable", "Printout"],
CellMargins -> {{18, Inherited}, {Inherited, Inherited}},
FontSize -> 10]}, Closed]]}, Open]],
Cell[
CellGroupData[{
Cell["Formulas and Programming", "Section"],
Cell[
CellGroupData[{
Cell[
StyleData["DisplayFormula"], CellMargins -> {{55, 10}, {2, 10}},
CellHorizontalScrolling -> True, DefaultFormatType ->
DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement" ->
True, HyphenationOptions -> {
"HyphenationCharacter" -> "\[Continuation]"}, LanguageCategory ->
"Formula", ScriptLevel -> 0, SingleLetterItalics -> True,
UnderoverscriptBoxOptions -> {LimitsPositioning -> True}],
Cell[
StyleData["DisplayFormula", "Presentation"],
CellMargins -> {{80, 10}, {5, 25}}, FontSize -> 18],
Cell[
StyleData["DisplayFormula", "Printout"], FontSize -> 10]},
Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["ChemicalFormula"], CellMargins -> {{55, 10}, {2, 10}},
DefaultFormatType -> DefaultInputFormatType,
"TwoByteSyntaxCharacterAutoReplacement" -> True,
HyphenationOptions -> {
"HyphenationCharacter" -> "\[Continuation]"}, LanguageCategory ->
"Formula", AutoSpacing -> False,
ScriptBaselineShifts -> {0.6, Automatic}, ScriptLevel -> 1,
SingleLetterItalics -> False, ZeroWidthTimes -> True],
Cell[
StyleData["ChemicalFormula", "Presentation"],
CellMargins -> {{80, 10}, {5, 15}}, FontSize -> 18],
Cell[
StyleData["ChemicalFormula", "Printout"], FontSize -> 10]},
Closed]],
Cell[
CellGroupData[{
Cell[
StyleData["Program"], CellMargins -> {{18, 10}, {Inherited, 6}},
Hyphenation -> False, LanguageCategory -> "Formula", FontFamily ->
"Courier"],
Cell[
StyleData["Program", "Presentation"],
CellMargins -> {{25, 10}, {8, 20}}, FontSize -> 16],
Cell[
StyleData["Program", "Printout"],
CellMargins -> {{18, 30}, {Inherited, 4}}, FontSize -> 9.5]},
Closed]]}, Closed]]}, Open]]}, Visible -> False, FrontEndVersion ->
"7.0 for Linux x86 (64-bit) (February 25, 2009)", StyleDefinitions ->
"Default.nb"]
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