Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay
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434 J. Hoelscher et al.
the ash content for all size fractions increases, then there will be a shift in the cut
point toward the outer section of the spiral. However, it is not known how much of a
shift will occur for a given flow rate and solid %. One way to answer these questions
would be to develop a model for the coal spiral for analysis.
Since the slurry is a two phase flow and current CFD techniques are unable to
model the flow easily, a comprehensive empirical model may be best suited for
the time being. In developing the model, the volumetric feed rate, solid %, and
the percent ash content would need to be varied. The model would also need to
have multiple measurements taken for each dependent variable in question. Further
guidelines for the model are as follows:
1. At the top of the spiral, the volumetric feed rate, solid %, and ash content should
be varied from high, medium, and low.
2. The ash content should be varied from high, medium, and low for all particle
sizes and not just one specific particle size.
3. At the bottom, the volumetric flow rate, pulp density, solid %, and % ash should
be measured or calculated for each section.
4. It is necessary to take multiple measurements to acquire a mean measurement
for each variable measured. (Three or more measurements should be taken.)
5. If an extensive project is conducted, other variables need to be considered
in order to be as thorough as possible to eliminate the need to go back and
conduct more trials. (For instance, taking measurements necessary to calculate
flow velocity along the spiral profile.)
6. Automation of the sampling and measuring may need to be implemented in order
to expedite and simplify the collection process.
After a good model for the coal spiral is developed, trends taking place within the
spiral may be able to be seen and actions can be taken to automate the spiral. Fur-
ther analysis may need to be done on-site at a coal preparation plant to determine
local trends in the spiral. This may be a better option since taking measurements
periodically from the bottom of the spiral and back calculating can give the same
results as varying the feed conditions in an experimental setup. If on-site analysis is
considered, measurements need only be taken from one spiral from one bank since
one bank is designed to provide the same flowrate and slurry mix to each of the
individual spirals.
33.5 Conclusions
Spiral concentrators have been widely used in coal preparation plants in Illinois
and elsewhere to clean 1 mm 150 m particle size coal fraction. The major
factors which have made spiral concentrator so popular include its low capital and
operating cost, no chemical reagent or dense medium requirement, and the ease of
operation/maintenance. Spirals are capable of providing excellent clean coal recov-
ery although at a relatively high ash content. Like any other water-only separation
33 Automating of Spiral Concentrators in Coal Preparation Plants 435
process, spirals are also susceptible to continuously fluctuating feed quality and
solids content in the feed slurry, which are quite common in a plant.
The main objective of the project is to develop an inexpensive control system for
spiral to automatically adjust the splitter position with fluctuating feed characteris-
tics to maintain the desired effective separation specific gravity (density cut-point).
The splitter actuation and the control system worked perfectly, but the sensing sys-
tem requires additional work. Two mechanical designs were investigated for the
sensing system, and both did not have the required resolution for effectively esti-
mating the specific gravity of the slurry across the profile of the spiral. Future work
entails investigation of other techniques such as ultrasonic, inductive and capacitive
based sensing.
An intriguing concept of using a model was also investigated, whereby it may be
possible to use simpler measurements, thereby circumventing the difficult problem
of estimating the specific gravity of the slurry in real time. Further investigation of
this approach also remains as future work.
Acknowledgments The authors would like to acknowledge the grant provided by ICCI
(# 06-1/2.1A.6) to support this work.
References
1. Abbot J (1982) The optimization of process parameters to maximize the profitability from a
three-component blend. In: Proceedings – 1st Australian coal preparation conference, April
6–10, New Castle, Australia, pp 87–105
2. Bethell PJ, Arnold BJ (2003) Comparing a two-stage spiral to two stages of spirals for fine
coal preparation. In: Honaker RQ, Forrest WR (eds) Symposium on recent advances in gravity
separation, SME Annual Meeting, Littleton, CO, pp 107–114
3. Luttrell GH, Kohmuench JN, Stanley FL, Trump GD (1999) An evaluation of multi-stage spiral
circuits. In: Proc. 16th International Coal Preparation Conference, pp 79–88.
4. Luttrell GH, Honaker RQ, Bethell PJ, Stanley FL (2003) Proceedings – 20th international coal
preparation exhibition and conference, Lexington, Kentucky, April 29–May 1, pp 69–78
5. Luttrell GH, Kohmuench JN, Stanley FL, Trump GD (1998) Improving spiral performance
using circuit analysis. Minerals and Metallurgical Processing Journal 15(4):16–21
6. Prinsolo TR, Abela RL (1998) Multiple stage fine coal spiral concentrators. In: Proceedings of
the international coal preparation congress, Brisbane, Australia, October 4–10, pp 245–253
7. Crowe CT, Elger DF, Roberson JA (2005) Engineering fluid mechanics, 8th edn. Wiley, New
York, p 441, Appendix A14
8. Yoon RH, Phillips DI, and Luttrell, GH., (2000) “Applications of Advanced Fine Coal Cleaning
and Dewatering Technologies,” Proceedings, 7th Asia-Pacific Economic Cooperation (APEC)
Clean Fossil Energy Technical Seminar, APEC Experts Group on Clean Fossil Energy, Chapter
VI (3), Taipei, China, March 7-8, 2000, 10 pp
Chapter 34
On the Rough Number Computation
and the Ada Language
Trong Wu
Abstract This chapter reports a numerical computation problem from theoretical
viewpoint. It shows computer systems are not capable for computation of real
numbers correctly due to the differences of algebraic structures between real num-
bers and model numbers. These two classes of numbers are not isomorphic. This
chapter defines several classes of real numbers such as model numbers, dyadic
numbers, finite dyadic numbers, limited dyadic numbers, and rough numbers, and
studies their structures. For numerical computation, this chapter proposes using the
concept of computer model numbers to approximate rough numbers for computa-
tion and selecting the Ada language for computation. The Ada language allows a
user to declare variables with appropriate number of digits or to specify a suitable
size of “small” for computation. In this way, the user can preset the desired accuracy
for all variables.
34.1 Introduction
In 1982, Pawlak proposed a concept called rough sets and used in the theory of
knowledge for the classification of features of objects [13]. He considered X to be
a set and R to be a relation over X. The pair .X; R/ is called a rough space,andR
is called the rough relation.IfxRy, then one could say that x is too close to y, x
and y are indiscernible, and x and y belong to the same elementary set. The concept
of rough sets is usually used in knowledge representation. Later, Wu defined a new
class of real numbers and called rough number [17, 19], which is one-dimensional
rough set in rough set theory.
Consider E be the set of all real numbers within the range of a given computer
system M ,andletR be a relation over E. The pair .E; R/ is called a rough space
and R is called the rough relation.Ifx;y 2 E and .x; y/ 2 R, we say that x
T. Wu (
)
Department of Computer Science, Southern Illinois University Edwardsville,
Edwardsville, IL 62026-1656, USA
e-mail: twu@siue.edu
A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,
DOI 10.1007/978-1-4419-5754-2
34,
c
Springer Science+Business Media, LLC 2010
437
438 T. Wu
and y are indistinctive in the rough space .E; R/ with respect to the given computer
system M . Wu called x and y rough numbers, and they are approximated by the
same model number [4,5], a number that the computer system can represent exactly.
In fact, if E is the set of all real numbers within the range of a given computer
system M ,andletR be a relation on E.Ifx, y 2 E and .x; y/ 2 R in the
rough space .E; R/ then R is an equivalence relation on E. Equivalence classes
of the relation R are called basic model intervals [15], the smallest interval with
model number endpoints. The set of all basic model intervals in .E; R/ is denoted
by E=R. The definition of a rough number is a one-dimensional rough set that is a
special case of rough sets givenbyPawlak[13].
The difficulty of numerical computation is that one must work with two distinct
number systems. Solving any numerical computation problem consists of the fol-
lowing three parts.
(1) The problem is given in the real number system.
(2) The computation is done in the model number system for the given machine, M .
(3) The results must be converted from model numbers into real numbers.
These two number systems have different algebraic structures, and they are not iso-
morphic. In general, moving from part (1) to part (2) can create errors and so can
moving from part (2) to part (3). This chapter reports the theoretical foundation
that a computer system is not capable of computing real numbers accurately within
its constraints. In order to manage the computation, we will introduce some basic
algebraic structures, the structures of dyadic numbers,andrough numbers. Then,
we will consider computing rough numbers with the Ada programming language.
34.2 Some Basic Algebraic Structures
For solving a numerical computation problem, there exists a fundamental crisis
in the algebraic structure of numbers. Different classes of numbers have different
algebraic structures. Therefore, we must study some basic algebraic structures of
numbers. In fact, the algebraic structures of real numbers, dyadic numbers,and
rough numbers have different algebraic structure. We will begin with the definition
of abelian group, then extend it to include a ring and a field [7]:
Definition 1. An abelian group (G; C)isasetG together with a binary operation
namely, addition, “C”, such that the following axioms are satisfied:
(1) For all a; b 2 G, such that a C b 2 G.
(2) For all a; b; c 2 G,then.a C b/ C c D a C .b C c/.
(3) There is an identity element, e in G such that a C e D e C a,foralla 2 G.
(4) For all a 2 G, there exists an inverse, .a/, such that .a/ C a
D a C .a/ D e.
(5) For all a; b 2 G, such that a C b D b C a.
34 On the Rough Number Computation and the Ada Language 439
Example 1. The set of all integers, I , with usual addition, C;.I;C/,isanabelian
group.Aring structure is a special case of an abelian group. By adding conditions
to an abelian group to form a ring:
Definition 2. We say that .SIC; / is a ring,if.SIC/ is an abelian group, defining
as a mapping from S S ! S, and satisfying the following axioms:
(1) For all a; b 2 S such that a C b D b C a.
(2) For all a; b; c 2 S ,then.a b/ c D a .b c/.
(3) Multiplication is distributed over addition: that is for all a; b; c 2 S,theleft
and right distributive laws hold:
a .b C c/ D a b C a c; and .b
C c/ a D b a C b c:
Example 2. The set of all integers I with usual addition + and multiplication ,
.I; C; / is a ring.Thisring has no zero-divisor.
A field is a commutative ring accompanied by unity with respect to multiplication
and every nonzero element has a multiplicative inverse.
Definition 3. A Field .F IC; / is a ring with commutative law and multiplicative
unity such that each nonzero element has a multiplicative inverse that satisfies the
following axioms:
(1) For all a; b 2 F such that a b D b a.
(2) There exists 1 such that a 1 D 1 a D a.
(3) For each nonzero element a in F , there is an inverse .1=a/ such that a.1=a/ D
.1=a/ a
D 1.
Example 3. The set of all rational numbers,real numbers,andcomplex numbers
with usual addition + and multiplication are fields.
Let F be a field,thesetofallpolynomials in the indeterminate, x, with co-
efficients in F and written as FŒx, then the following theorem (without proof)
determines a ring of polynomials [7]:
Theorem 1. Let F be a field, the set of all polynomials a
0
Ca
1
xCa
2
x
2
CCa
n
x
n
,
written as FŒx,wheren can be any nonnegative integer and a
i
.i D 0;1;:::;n/
are all in F . Then, FŒx is a ring under the operation induced by the opera-
tions in F .
It is known that the set of all real numbers, denoted R, together with arithmetic
addition and multiplication forms a field. Human arithmetic, C and , always re-
turns exact results. On the other hand, most computer systems can store only certain
real numbers exactly in the memory. Some of these real numbers are 100.5, 70.875,
12.375, 0.5, 87.125, etc. Some other real numbers such as 0.1, 0.2, 0.3, 0.4, 0.6, etc.
cannot be stored exactly in the memory. This tells us that the computer system does
not actually have these real numbers. Therefore, the set of all numbers represented
by the computer for computation with the usual addition, C, and multiplication,
operations are not the same as the set of all real numbers. Even if we allow a com-
puter system to have as many bits as required to store a floating-point number;it
is still not able to have their multiplicative inverse for all numbers in the computer
440 T. Wu
system. Because of these, computer numbers do not form a field. The structures of
the set of all real numbers and the set of all the numbers in the computer system are
quite different. They are not isomorphic. This leads to a computation of C and
over the set of all numbers represented by a computer system that can induce some
errors. Therefore, we need to find out the actual structure of computer-represented
numbers. What kind of arithmetic can computer systems do? So, we will now turn
to study the set of dyadic numbers [10].
34.3 The Structure of Dyadic Numbers
We will require that for each nonnegative real number x a b-adic expansion,whereb
is an integer greater than 1. Actually, we want to write a real number x as the sum of
multiples of powers of b, where the multiples are nonnegative integers less than b.
Clearly, the b-adic expansion of the number may fail to be unique in a decimal
expansion, 0:9999 : : : (all nines) and 1.0000. .. (all zeros) are to be expansions of
the same real number.Whenb D 2,theb-adic expansions are then called dyadic
expansions and numbers written as dyadic expansions are called dyadic numbers
[10].
Theorem 2. For each real number x, we have its dyadic expansion over a finite
field (BIC; ):
x D sgn
n
X
iDn
a
i
2
i
; where a
i
2 B Df0; 1gand sgn 2fC; g: (34.1)
A computer system is a finite state machine; therefore, it is capable of representing
only a finite set of numbers internally. Thus, any attempt to use a digital computer to
do arithmetic in the set of all real numbers is doomed to failure. The set of all real
numbers is an infinite set, and most of the elements in the set cannot be represented
by a computer system. Therefore, for theoretical reasons, we may assume that a
computer system can have any finite number of bits to store its numbers, integers
and floating-point numbers.
For any nonnegative integer n, we consider the set of all numbers with the repre-
sentation:
y D sgn
n
X
iDn
a
i
2
i
;a
i
2 B; sgn 2fC; g: (34.2)
This representation contains two parts, one is with index 0 i n and the other
is with an index 1 i n. Actually, these two parts are the integer part of
y and fraction part of y, respectively. In this, we define a structure for the set of
all numbers with the representation, denoted D and call the set of all finite (term)
dyadic numbers,andthetermoffinite means any finite value:
D D
(
f
y
j
y D sgn
n
X
iDn
a
i
2
i
;a
i
2 B; sgn 2fC; g;n2f0; Z
C
g
)
:
(34.3)
34 On the Rough Number Computation and the Ada Language 441
From Definitions2, 3, and 2.4, we see that all finite (term) dyadic numbers D
comprise neither a ring nor a field. More over, it is not a ring of polynomials of
2 over a finite field B D .f0; 1gIC; /.
In fact, each computer system can have only a fixed number of bits of memory
space to store a number of its type such as integer, float, double ;:::;etc. An integer
within the computer-predefined range often can be represented exactly. However, a
real number within the range usually cannot be represented correctly. Today, most
computer systems implement the IEEE floating-point number format [9] for storing
real numbers. For example, a 32-bit floating-point number format is divided into
three areas: a sign bit sign, an 8-bit exponent E, and a 23-bit mantissa M : sign
.1 M/ 2
E127
,wheresign D ‘0’ indicates a positive value, sign D‘1’ indicates
a negative value. The 1 in the .1 M/is the hidden bit, M is a 23-bit mantissa, and
exponent E is in f0; 1; 2; : : :255g. It is clear that a floating-point number representa-
tion can be given in the set of D in (34.3). Again, computer systems can only use a
limited number of bits to represent a floating-number. Therefore, ring computation,
C and , can cause an overflow or underflow. Most computer systems use floating-
point number arithmetic for numerical computation, and most computer users are
not aware that floating-point number arithmetic often can create rounding-errors
during the course of a computation. This error is unavoidable. In general, the output
of a numerical computation program from one machine can be different from the
output from another machine.
34.4 The Structure of Rough Numbers
The Ada programming language [5, 6, 15] calls a real number a model number,if
the real number can be stored exactly in a computer system with a radix of power
of 2. Therefore, the set of all model numbers is a subset of dyadic numbers,andwe
callitthesetoflimited dyadic numbers with respect to a given computer system.
The term limited reflects the given computer system’s word size for floating-point
numbers. For those real numbers that cannot be stored exactly in a given computer
system, we call them rough numbers with respect to the given computer system. In
order to study this large class of numbers, it is necessary to define rough numbers
precisely and mathematically.
Definition 4. Let E be the set of all real numbers within the range of a given com-
puter system M ,andletR be a relation on E. The pair .E; R/ is called rough space
and R is called the rough relation.Ifx; y 2 E and .x; y/ 2 R, we say that x and y
are indistinctive in the rough space .E; R/ with respect to the given computer system
M . We call x and y rough numbers, and they are approximated to the same model
number.Ifareal number is a model number with respect to a given machine M ,
then it is a special case rough number. Most computer systems use a downward
rounding policy to approximate a rough number with a model number. Thus, there
are infinitely many rough numbers approximated by the same model number,such
that all
rough numbers z 2 Œx; y/ are approximated by the same value of the model
442 T. Wu
number x for computation. Throughout this chapter, we use downward rounding
policy to handle the approximation of rough numbers [15,17,19].
Theorem 3. Let E be the set of all real numbers within the range of a given com-
puter system M , and let R bearelationonE.Ifx;y 2 E and .x; y/ 2 R are in the
rough space .E; R/,thenR is an equivalence relation on E (Readers may verify
this theorem).
Equivalence classes of the relation R are called basic model intervals [15], the
smallest interval with model number endpoints. The set of all basic model inter-
vals in .E; R/ is denoted by E=R. The definition of a rough number is a special
case of the one-dimensional rough sets given by Pawlak [13].
In reality, for any rough number z whose value is within the range of a given
computer system M , there exists the smallest closed-open interval, such that
z 2
i C
k 1
2
n
;i C
k
2
n
for some positive integersn and k; (34.4)
where the lower bound is the approximation of z;i is the integer part of z;k is an
integer with 0 k 2
n
1,andn D n.b
e
;b
m
/, a function of the number of bits in
the field of exponential b
e
, and the number of bits in the field of mantissa b
m
.
The difference, d , between the rough number z, and its approximation given in
(34.4)isgivenby
d D z
i C
k 1
2
n
: (34.5)
This is also called rounding-error. Usually, a programmer is either unaware of the
existence of rounding-errors or unable to reduce or control them in the course of
a computation. We will show that the set of all rough numbers over the set of all
computer systems has the dyadic number structure.
Most computer users are not aware that the computation of real numbers and
rough numbers are not the same. Even more confusing, some programming lan-
guages such as FORTRAN, COBOL, Pascal, etc. allow programmers to declare
variables with type real in their programs for computations. To make it clear to all
computer users, we must study the algebraic structures for the set of all real numbers
and the set of all rough numbers over computer systems. This not only provides the
theoretical foundation for computer arithmetic, but also interprets rounding-errors
from an algebraic theoretical viewpoint.
34.5 Hierarchical Classes and Computation
In mathematics, we have learned properties of several classes of numbers such
as integers, rational numbers, irrational numbers,andreal numbers.However,a
computer system can have only a limited word size; therefore, only some inte-
gers and some rational numbers can be computed correctly in a computer system.
34 On the Rough Number Computation and the Ada Language 443
Real Number
Dyadic Numbers
f
a,b
a
+
b
()fa b
+
(), ()fa fb
Rough numbers
Limited Dyadic Numbers
(Model numbers)
Finite Dyadic Numbers
Fig. 34.1 Hierarchical structures of dyadic numbers
Other numbers cannot even be represented exactly in a computer system. Computer
systems handle real numbers in a binary fashion. For each real number x,wehave
its dyadic expansion over the finite field .BIC; /. Among all the dyadic numbers,
only a limited number of dyadic numbers can be computed in a computer system.
The relationship of the set of real numbers and the hierarchical structures of dyadic
numbers is shown in Fig. 34.1.Onthesetofreal numbers, we perform a field com-
putation, which is an exact computation. However, when a real number within the
given range is stored or read into a computer system, it is converted into a dyadic
number. The computation over the set of limited dyadic numbers is the dyadic num-
ber computation. Let f be a mapping that takes each real number into its dyadic
representation. Consider real numbers a and b within a given range mapped into
their dyadic representations f.a/ and f.b/respectively.
In general, we should have
f.aC b/ ¤ f.a/˚ f.b/: (34.6)
The set of real numbers has a field structure, while the set of finite dyadic numbers
is not a field. f does not preserve the structure, and f cannot even be a local homo-
morphism between the set of real numbers within a given range of machine M and
the set of limited dyadic numbers. The addition ‘C’ (or multiplication ‘’) on the
right hand side of (
34.6)isafield addition, and the addition ‘˚’ on the left hand side
is a ring addition. They are two different additions. The ‘˚’isthedyadic number
addition. To adjust the inequality (34.6), we will add in an error term, Err:
f.aC b/ ¤ f.a/˚ f.b/C Err: (34.7)
This error term, Err, is called rounding-error.
Unfortunately, most programming textbooks do not teach students the addition
‘C’ in program is not real number addition, but dyadic number addition instead.
Many computer users are misled and disappointed that computer systems cannot
provide accurate results for computations.
444 T. Wu
Among all commonly used programming languages, the Ada programming
language is a unique language that allows the user to define a specific model number
for their rough numbers for dealing with computation. In the next section, we will
introduce the Ada language briefly. Then, we will study its special feature in the
defining model numbers for the computation of rough numbers.
34.6 The Ada Language and Rough Number Computation
In the 1970s, the US Department of Defense (DoD) realized that it was spending too
much for its software, particularly in its real-time or embedded systems.TheDoD
found that they used several hundred special-purpose programminglanguages for its
real-time systems. The DoD decided to design a new language to replace all of these
languages, and this new language would be a general-purpose language as well. Fol-
lowing a number of definitions of requirements, developments, and evaluations, the
ANSI standard for the Ada language issued in 1983 [5,6]. The document containing
the final requirements for the Ada 83 language was called Steelman. The team that
developed the language was CII Honeywell Bull in France, and the leader of the
design team was Jean D. Ichbiah. The language was named Ada after Augusta Ada
Byron, Countess of Lovelace (1815–1852), and the daughter of Lord Byron. She
worked as an assistant for Charles Babbage’s mechanical analysis engine. It was
very well believed that she was the world’s first programmer.
After the Ada language had several years of real-world application and users
learned object-oriented programming techniques, the DoD established the Ada 9X
revision project in 1988. It was based on the original Ada 83, but included additions
such as type extensions, hierarchical libraries, a greater ability to manipulate points
or reference types, object-oriented programming without incurring overhead, and a
tasking model. Ada 9X remained very strongly type checking. Later, this revised
language was standardized and named the Ada 95 language [2].
The Ada language’s real number types are subdivided into float-point types
and fixed-point types.Inthefloat-point type, values are numbers with the format
˙:dd :::d 10
˙dd
.Forfixed-point types, values are numbers with the formats
˙dd:ddd; ˙dddd:0 or ˙ 0:00ddd [1, 3, 18]. From Definition 4, we have learned
that all rough numbers z 2 Œx; y/ are approximated by the same value of the model
number x for computation. For the float-point number types,amodel number,other
than zero that can be represented exactly by a given computer, is of the form:
sign mantissa .radix/
.exponent/
: (34.8)
In this form, sign is either C1 or 1; mantissa is expressed in a number base given
by radix and exponent is an integer. The Ada language allows the user to specify
or to declare the number of significant decimal digits needed. This is downward
rounding of a rough number to its approximated model number.Afloating-point
type declaration with or without the optional range constraint is shown:
Type T is digit DŒrange L R: (34.9)