\documentclass{amsart}
\usepackage{amssymb,amsfonts,latexsym,amsmath}

\begin{document}

\title{References, Index, \& Symbols and Notations}
\maketitle

\section*{References}

Amari, S. (1972) Learning patterns and pattern
sequences by self-organizing nets of threshold elements. IEEE
Trans. Comput. 21, 1197-1206.

Amit, D. J. (1989) Modeling Brain Function - The world of
attractor neural networks (Cambridge University,Cambridge).\\
 
Arimoto, S. (1963) Periodic sequences of states of an
autonomous circuit consisting of threshold elements.
Trans. Inst. Electron. Comm. Eng., Studies on Information \&
Control 2 (in Japanese), 17-22.

Biggs, N. L. (1989) Discrete Mathematics (Oxford        
University, New York).
 
Blum, E. K. (1990) Mathematical aspects of outer-product
asynchronous content-addressable memories. Biol.
Cybern. 62, 337-348.

Bressloff, P. C. and Taylor, J. G. (1991) Discrete time    
leaky integrator network with synaptic noise. Neural
Networks 4, 789-801.

Chow, C. K. (1961) Boolean functions realizable     with
single threshold devices, Proc. IRE. 49, 370-371.

Coombes, S., Doole, S. H. (1996) Neural population
dynamics with post inhibitory rebound: A reduction to
piecewise linear discontinuous circle maps. Dynamics and
Stability of Systems 11, 193-217. 

Cottrell, M. (1988) Stability and attractivity in    
associative memory networks, Biol. Cybern. 58, 129-139.

Elgot, C. C. (1961) Truth functions realizable by
single threshold organs. SCLTD, 225-245.
Goles-Chacc, E. (1980) Comportment oscillatoire d'une    
famille d'automates cellulaires, non uniformes. Th\`{e}se
IMAG, Grenoble.

Goles, E. and Olivos, J. (1981) Comportment periodique des
fonctions a seuil binaires et applications. Discrete Appl.
Math. 3 93-105.

        Goles, E., Fogelman-Soulie, F. and Pellegrin, D.
(1985) Decreasing energy functions as a tool for studying
threshold networks. Discrete Appl. Math. 12, 261-277.
Gray, F. (1953) Pulse code communication. U.S. Patent 2    
632 058, March 17.

Grove, L. C. and Benson, C. T. (1985) Finite Reflection    
Groups (2nd ed.) (Springer, New York). 
Hall, M. (1967) Combinatorial Theory (Blaisdell,        
Waltham, MA).

        Hopfield, J. J. (1982) Neural networks and physical
systems with emergent collective computational abilities.
Proc. Nat. Acad. Sci. USA 79, 2554-2558. 

Hu, S. T. (1965) Threshold Logic (University of        
California, Berkeley).

Ishii, N. (1970) On transformation between state       
transition diagrams realizable by the network of     threshold
elements. J. Electron. Comm. Eng. 53-C-    10, 715-723 (in
Japanese). 

Ishii, N. and Miyazaki, K. (1972) Generation of a set     of
periodic sequences realizable by the network of     threshold
elements. J. Electron. Comm. Eng. 55-D-8,     491-498 (in
Japanese).

Kalat, J. W. (1995) Biological Psychology (5th ed.)    
(Brooks/Cole, CA).

Kamp, Y. and Hasler, M. (1990) Recursive Neural        
Networks for Associative Memory (John Wiley, New     York). 

Krishnamurthy, V. (1986) Combinatorics-Theory and
Applications (Ellis Horwood, Chichester). 

McCulloch, W. S. and Pitts, W. H. (1943) A logical
calculus of the ideas immanent in nervous activity. Bull.
Math. Biophysics 5, 115-133. Reprinted in J. A. Anderson and
E. Rosenfield, ed, (1989) Neurocomputing: foundations of
research (MIT, Cambridge) 18-31.

Masters G. M. and Mattson, R. L. (1966) The
application of threshold logic to the design of sequential
machines. IEEE Symposium on Switching and Automata Theory,
184-194.

Mathematical Society of Japan (1987) Encyclopedic    
Dictionary of Mathematics (2nd English ed.) (MIT,    
Cambridge, MA).

Muroga, S. (1971) Threshold Logic and Its Applications
(Wiley-Interscience, New York)

Muroga, S., Toda, I. and Takasu, S. (1961) Theory of
majority decision elements. J. Franklin Inst. 271, 376-418.
Originally in J. Inst. Electron. Comm. Eng. Japan 43 (1960). 

Ore, O. (1988) Number Theory and its History (Dover, New
York).
 
Robert, F. (1986) Discrete Iterations (Springer,        
Berlin).

Seung, H.S., Lee, D.D., Reis B.Y., and Tank. D. W. (2000)
Stability of the memory of eye position in a recurrent network of
conductance-based model 
neurons. Neuron 26, 259-271.

Ueda, T. (1977) The fixing groups for the 2-asummable
Boolean functions. Discrete Math. 20, 77-82.

Ueda, T. (1979) On the Fixing group for a totally pre-
ordered Boolean function. Discrete Math. 26, 293-295.

Ueda, T. (1992) Circular non-singular threshold
transformations. Discrete Math. 105, 249-258. 

Ueda, T. (1994) Graphs of non-singular threshold
transformations. Discrete Math. 128, 349-359.

Ueda, T. (2000) Reflectiveness and compression of    
threshold transformations. Discrete Appl. Math. 107,
215-224.

Ueda, T. (2001) An enhanced Arimoto theorem on        
threshold transformations. Graphs and Combinatorics
17, 343-351.

Williamson, S. G. (1985) Combinatorics for Computer
Science. Rockville, MD: Computer Science Press.

Winder, R. O. (1962) Threshold logic. Ph.D. Thesis,    
Mathematics Department, Princeton University.

Yajima, S. and Ibaraki, T. (1968) A theory of        
completely monotonic functions and its applications     to
threshold logic, IEEE Trans. Comput. 17, 214-229.

  
  
  \pagebreak
  
 \section*{INDEX}


  
abelian group \quad 1.2

action \quad 1.3

action potential \quad   6.2

AND \quad   1.4

Arimoto

\qquad  $\sim$'s theorem \quad 5.5

\qquad  enhanced $\sim$  theorem \quad 5.5

\qquad  -Ueda theorem \quad  6.2  

asummable \quad 4.2

\qquad   m-asummable \quad   4.2

attractive \quad  6.1, 9.2

attractor \quad 6.1, 9.2

\qquad  connected $\sim$ \quad  6.1

\qquad  Cottrell's $k$-$\sim$ \quad  6.2

\qquad  minimal $\sim$   \quad 6.1

\qquad  separated $\sim$ \quad  6.1

\qquad  strong $\sim$ \quad  6.1

basin for an attractor \quad 6.1, 9.2

bi-dependent \quad 9.3

bijection \quad 1.1

Boolean  

\qquad  function \quad 1.4

\qquad  isometry \quad 2.1

\qquad  minimal $\sim$ algebra \quad   1.4

\qquad  transformation \quad 1.4

burst neuron \quad 10.1

carrier \quad 1.1

Cartesian product \quad  1.1

circular

\qquad  transformation \quad  2.4

\qquad   skew-$\sim$ transformation \quad 2.4

complement \quad  1.1, 2.1

complementation \quad 1.4

complete set \quad  2.3

completely monotonic \quad 4.2

composition \quad 1.1

\qquad  disjoint $\sim$   \quad  1.1  

compressible \quad 5.4

compression \quad 5.2

conjugate \quad 1.2

conjunction \quad 1.4

congruent \quad 1.2

copies \quad 3.5

cube

\qquad  $n$-dimensional $\sim$ \quad  2.1

\qquad  $n$-cube \quad 2.1

cycle  

\qquad   Hamiltonian $\sim$ \quad 1.1

\qquad  $k$-cycle \quad 1.1, 9.2

\qquad  limit $\sim$ \quad  6.2

\qquad  of a digraph \quad  1.1

\qquad  of a graph \quad  1.1

\qquad  of $\mathrm{GRAPH}(F)$  1.1

\qquad  neutral $\sim$ \quad 6.3

cycle structure \quad  2.3

cyclic

\qquad  $\sim$  orbit \quad 6.1, 9.2 

\qquad  $\sim$  permutation \quad 1.2

\qquad  $\sim$  sequence \quad 6.1

De Morgan's law \quad 1.4

density \quad 1.3 

digraph \quad 1.1

dimension 

\qquad of DNN \quad 6.3 

direct product

\qquad  of groups \quad 1.2 

\qquad  of sets \quad 1.1

\qquad  of transformations \quad 1.1

direct sum \quad 1.1

disjoint composition \quad 1.1

disjunction \quad 1.4

disjunctive form \quad 1.4

\qquad  irredundant $\sim$ \quad  1.4

distance \quad 1.1

\qquad  Euclidean $\sim$ \quad  2.2

\qquad  Hamming $\sim$ \quad 2.1

DNN (dynamical neural network) \quad  6.3

dynamical neural network (DNN) \quad 6.3

dynamical system

\qquad  finite-state $\sim$   \quad  6.1, 9.2

efficacy of a synapse \quad 6.2

elementary self-dual transformation \quad 2.3

Elgot's theorem \quad 4.2

equivalent

\qquad  Boolean transformations \quad 2.1

\qquad  isometrically $\sim$ \quad  2.2

\qquad  orthogonally $\sim$ \quad  2.2

Euclidean distance \quad 2.2

Euler's function \quad 1.2

expansible

\qquad  threshold transformations \quad 5.2

expansion \quad 5.2

extended representation \quad 7.1, 8.4

face copies  \quad 3.5

finite-state dynamical system (FSDS) \quad 6.1

\qquad   parametrized $\sim$ \quad 6.1

firing rate \quad 6.3

\qquad  maximum $\sim$ \quad 6.3

\qquad  minimum $\sim$ \quad 6.3

\qquad  spontaneous $\sim$ \quad 6.3

fixed point 

\qquad  of a transformation \quad  1.1

flow

\qquad  of a transformation \quad  2.5 

flow graph \quad 2.5

FSDS (finite-state dynamical system)  \quad  6.1

\qquad  parametrized $\sim$ \quad  6.1

Goles-Chacc's theorem  \quad  6.2

Gray code

\qquad  binary-reflected $\sim$  \quad  3.1

graph  \quad  1.1

\qquad  directed $\sim$  \quad  1.1

\qquad  of a transformation  \quad  1.1

Hamming distance  \quad  2.1

homomorphism  

\qquad  for groups  \quad  1.2

Hopfield model  \quad  6.2

image

\qquad of a function \quad 1.1

\qquad of a sequence \quad 1.1
 
\qquad of a set of sequences \quad 6.1

imaginary

\qquad  delay neuron \quad 8.1

\qquad   synapse  \quad  6.3

implicant  \quad  1.4

\qquad   prime $\sim$  \quad  1.4

injection  \quad  1.1

incompressible  \quad  5.2

inexpansible  \quad  5.2

invertible  \quad  1.2

incompressible

\qquad   transformations  \quad  5.2

inexpansible

\qquad  threshold transformations  \quad  5.2

invariant 

\qquad   set  \quad  6.1

irredundant

\qquad  disjunctive form  \quad  1.4

isometrically equivalent \quad  2.2

isometrically similar  \quad  2.2

isometry  \quad  2.1

isomorphic graphs  \quad  1.1

isomorphism of graphs  \quad  1.1

length

\qquad  of a path  \quad  1.1

\qquad  of a cycle  \quad  1.1

limit cycle  \quad  6.1

limit orbit \quad  9.2

limit set  \quad  6.1

linear permutation  \quad  1.2

linear separability  \quad  4.2

loop  \quad  1.1

McCulloch-Pitts model  \quad  6.2

Maximal

\qquad  transformation  \quad  2.2

\qquad  uniquely $\sim$  transformation  \quad  2.2

metric space  \quad  1.1

minimal

\qquad  attractor  \quad  6.1

\qquad  Boolean algebra  \quad  1.4

\qquad  transformation  \quad  2.2

\qquad  uniquely $\sim$  transformation  \quad  2.2  

modulo

\qquad  congruent $\sim$  \quad  1.2

monotonic

\qquad  completely $\sim$  \quad  4.2

\qquad  $k$-monotonic  \quad  4.2

$n$-cube  \quad  2.1

negation  \quad  1.4  

neural network

\qquad  attractor $\sim$ \quad  iii

\qquad  feed forward $\sim$ \quad  iii

\qquad  dynamical $\sim$  (DNN)  \quad  6.3

\qquad  recursive $\sim$ \quad  iii

neutral cycle  \quad  6.3

OR  1.4

orbit

\qquad  of an action group \quad  1.3,

\qquad  in a FSDS \quad  6.1, 9.2

\qquad  limit $\sim$ \quad  9.2

\qquad  periodic $\sim$ \quad  9.2

orbit modification  \quad  3.5

\qquad  Arimoto's $\sim$  \quad  5.5

order of a group \quad  1.2

orthogonal transformation \quad  2.1

orthogonally equivalent

\qquad  transformation \quad  2.2

orthogonally similar

\qquad  transformation \quad  2.2

parallel  \quad  6.2

partition  \quad 1.1

path  \quad  1.1

\qquad  Hamiltonian $\sim$   \quad  1.1

PDNN \quad 6.3, 8.1 

period  \quad  1.3, 9.2

periodic

\qquad  $\sim$  orbit \quad 9.2

\qquad  $\sim$  sequence \quad 9.2

permutation  \quad  1.2

\qquad  cyclic $\sim$   \quad  1.2

\qquad  linear $\sim$   \quad  1.2

\qquad  of coordinates  \quad  1.3

P\'{o}lya action  \quad  1.3

postsynaptic

\qquad  neuron  \quad  6.2    

potential

\qquad  action $\sim$  \quad  6.2

\qquad  postsynaptic $\sim$  \quad  6.2

preorder  \quad  4.2

presynaptic neuron  \quad  6.2

prime

\qquad  implicant  \quad  1.4

primitive DNN  \quad  6.3, 8.1

prototype DNN  \quad  6.3, 8.1

realizable

\qquad  simultaneously $\sim$  \quad  4.1

realize  

\qquad  a cycle structure  \quad  2.3

real neuron \quad 8.1

reflection subgroup  \quad  2.1

reflective transformation \quad  3.1

representation

\qquad $\sim$  of a group  \quad  1.3

\qquad [\;]-$\sim$  \quad  2.3

\qquad extended $\sim$   7.1, 8.4

residue class ring  \quad  1.2

rotation

\qquad  right $\sim$  \quad  1.3


runs-based distance \quad 7.7

runs pair \quad 7.7

segment

\qquad  of a linear permutation  \quad  1.2

self-dual

\qquad  function  \quad  2.3

\qquad  transformation  \quad  2.3

separated

\qquad  by a hyperplane  \quad  4.1

\qquad  attractors  \quad  6.1

sequence

\qquad  cyclic \quad 9.2

\qquad  $k$-sequence \quad 9.2

\qquad  periodic \quad 9.2

serial  \quad  6.2

similar

\qquad  isometrically $\sim$  \quad  2.2

\qquad  orthogonally $\sim$  \quad  2.2

simultaneously realizable  \quad  4.1

skew-circular

\qquad  transformation  \quad  2.4

slope  

\qquad  of a linear permutation \quad 1.2 

spatial summation \quad 8.1

spontaneous

\qquad  firing  \quad  6.3, 8.1

\qquad  firing rate  \quad  6.3, 8.1

stabilizer  \quad  1.3

stable

\qquad  in Amari's sense  \quad  6.2

\qquad  structurally $\sim$  \quad  6.1

states  \quad  6.1    

state space  \quad  6.1  

structurally stable \quad  6.1

sum

\qquad  of transformations \quad  1.1

summable  \quad  4.2

\qquad  $m$-summable  \quad  4.2

surjection  \quad  1.1

symmetric

\qquad  difference  \quad  1.1

\qquad  group  \quad  1.2

synapse  \quad  6.2

synchronization  \quad  6.3 

temporal summation  \quad  6.3

term  \quad  1.4

threshold

\qquad  function  \quad  4.1

\qquad transformation  \quad  4.1  

\qquad  value  \quad  4.1 

\qquad  vector  \quad  4.3

tonic  \quad  10.2

total preorder  \quad  4.2

transformation  \quad  1.1

\qquad  group  \quad  1.3

\qquad  one-to-one $\sim$  \quad  1.1

2-asummable  \quad  4.2

unate  \quad  4.2

variation  \quad  2.2

walk  \quad  1.1

weight

\qquad  vector  \quad  4.1

\qquad  matrix  \quad  4.3

wreath product  \quad  2.1

Yajima-Ibaraki theorem  \quad  4.2

  

  \pagebreak


  \section*{Symbols and notations}

$$
\begin{array}{ll}
1_{A} &   \mbox{Characteristic function for subset $A$} \quad
1.4\\
\mathbf{B}  & \mbox{Set of Boolean functions: $\mathbf{Q}^{n}
\rightarrow  \mathbf{Q}$} \quad 2.1\\
Bool   & \mbox{Function: $\mathbf{R}^{n} \rightarrow 
\mathbf{Q}^{n}$} \quad 2.1\\
\mathrm{Car}F & \mbox{Carrier of transformation $F$}
\quad 1.1\\
\mathrm{CS}(F) &  \mbox{Cycle structure of transformation $F$}
\quad 2.3\\
\mathrm{CY}(F) & \mbox{Set of all cycles in $\mathrm{GRAPH}(F)$}
\quad 1.1\\
d(x) & \mbox{Density of $x$} \quad 1.3\\
d(x,y) &   \mbox{Distance} \quad 1.1\\
d(x,A) &  Distance between a point and sets 1.1\\
d(A,B) &  Distance between sets  1.1\\
d_{H}(x,y) & \mbox{Hamming distance} \quad 2.1\\
d_{R}(x,y) & \mbox{Runs-based distance} \quad 7.4\\
f|a & \mbox{Boolean function defined by $(f|a)x =
f(x,a)$} \quad 1.4\\
\mathrm{gcd} & \mbox{Greatest common devisor} \quad 1.2\\
\mathrm{GRAPH}(F) & \mbox{Graph of transformation $F$} \quad
1.1\\
I &  \mbox{Identity transformation} \quad 2.1\\
\mathrm{Im}\Psi  & \mbox{Image of a set $\Psi $ of sequences}
\quad 6.1\\
l &  \mbox{Column vector whose every element is 1} \quad  4.3\\
\mathbf{N}_m &   \mbox{Residue class ring with $m$ elements $\{
1,2,...,m\}$} \quad 1.2\\
1^{m} &  m-\mbox{vector whose every coordinate is 1} \quad 5.5,
7.1\\
0^{m} &  m-\mbox{vector whose every coordinate is 0} \quad  5.5,
7.1\\
\mathbf{N}   &    \mbox{Residue class ring with n elements $\{
1,2,...,n\}$, i.e. $\mathbf{N}_n$} \quad 1.2\\
O(\mathbf{Q}^{n}) & \mbox{Group of Boolean isometries} \quad
2.1\\
O(\{-1, 1\}^{n}) & \mbox{Group of orthogonal transformations of
$\{ -1,1\} ^{n}$} \quad 2.1\\
O(\mathbf{R}^{n}) & \mbox{Real orthogonal group} \quad 2.1 \\
o    &    \mbox{Column vector whose every coordinate is 0} \quad
6.2\\
\mathrm{Orb}_{\mathbf{G}}x &   \mbox{Orbit of $\mathbf{G}$ acting
on $X$ containing $x \in  X$} \quad 1.3\\
\mathrm{Orb}_{\mathbf{G}}S & \mbox{Union of orbits
$\mathrm{Orb}_{\mathbf{G}}x$ for $x \in S$} \quad  1.3\\
\mathrm{Orb}_{F}x & \mbox{Orbit starting at $x$ in a
FSDS generated by $F$} \quad 6.1\\
\mathrm{Orb}_{F}S & \mbox{Set of orbits starting at some $x \in 
S$ in a FSDS generated by $F$} \quad 6.1\\
\mathrm{Orb}_{F,V}x & \mbox{Orbit starting at $x$ in a
non-autonomous FSDS generated by $F$ and $V$} \quad 9.2\\
\mathrm{Orb}_{F,V}S & \mbox{Set of orbits starting at some $x \in 
S$}\quad\\  & \mbox{in a  on-autonomous FSDS generated by $F$ and
$V$} \quad 9.1\\
\mathrm{PO}(F,V)  & \mbox{Set of all periodic orbits in the
non-autonomous} \quad\\
 & \mbox{FSDS generated by $F$ and $V$}  \quad 9.2\\
P_{L} & \mbox{Projection function: $\mathbf{Q}^{M} \rightarrow 
\mathbf{Q}^{L}$} \quad 1.4\\
p_{i} & \mbox{Projection function: $\mathbf{Q}^{n} \rightarrow 
\mathbf{Q}$ defined by $p_{i}x = x_{i}$} \quad 1.4\\
\mathbf{Q} & \mbox{Minimal Boolean algebra $\{ 0,1\}$} \quad  
1.3, 1.4\\
\mathbf{Q}^{n} & \mbox{Set of all functions from $\mathbf{N}$ to
$\mathbf{Q}$,}\quad\\
  & \mbox{i.e. $n$-bit binary strings $x_{1}\cdot x_{2}\cdot
\cdot \cdot x_{n}$} \quad 1.3\\

\mathbf{R}^{n} & \mbox{Real $n$-dimensional space} \quad 2.1\\
\mathrm{RP} & \mbox{Number of non-overlapping runs pairs}
\quad 7.4\\  
S_{m}{ \cdot  } & \mbox{The disjunction of all conjunctions of
$m$ Boolean}\quad \\
 & \mbox{functions or variables selected from $\{ \cdot  \}$}
\quad 4.1\\
\mathrm{SCY}(F) & \mbox{Set of all significant cycles} \quad
6.3\\
\mathrm{Sgn} & \mbox{Function: $\mathbf{R}^{n} \rightarrow  \{
-1,1\}^{n}$} \quad 2.1\\
\end{array}
$$

$$
\begin{array}{ll}
\mathrm{SYM}(X) & \mbox{Symmetric group on set $X$} \quad 1.2\\
\mathbf{U}_{n}  & \mbox{Multiplicative abelian group} \quad
1.2\\
U_{\epsilon}S  &  \epsilon \mbox{-neighborhood of a subset }S
\quad 6.1\\ 
\mathbf{V}     & Image of a sequence V   1.1\\
\mathrm{Var}(F) &   \mbox{Variation of transformation $F$} \quad
2.2\\
X^{D}   &   \mbox{Set of all functions from $D$ to $X$} \quad
1.1\\
\mathbf{Z} & \mbox{Set of all integers} \quad 1.1\\
\mathbf{Z}_{n} & \mbox{Residue class ring with $n$ elements $\{
0,1, ...,n-1\}$} \quad 1.2\\
\mathbf{Z}_{+} & \mbox{Set of non-negative integers $\{
0,1,2,...\}$}  \quad 1.1\\
\mathbf{Z}^{+} & \mbox{Set of positive integers $\{
1,2,...\}$}  \quad 1.1\\
\iota    &     \mbox{Identity permutation} \quad 1.2\\
\rho     &    \mbox{Circular permutation (1,2,..,n)} \quad 1.2\\
\varphi (n) &   \mbox{Euler's function} \quad 1.2\\
\omega _{F}x   & \mbox{Limit set of $x$ in the FSDS generated by
$F$} \quad  6.1\\
\omega _{F}A & \mbox{Union of limit sets of some $x \in  A$ in
the FSDS generated by $F$} \quad 6.1\\
\Omega _{F,V}x &   \mbox{Limit orbit of $x$ in the non-autonomous
FSDS generated by $F$ and $V$} \quad 9.2\\
\Omega _{F,V}A &    \mbox{Union of limit orbits of some $x \in 
A$} \quad\\
& \mbox{in the FSDS generated by $F$ and $V$} \quad 9.2\\
|X| & \mbox{Number of elements of finite set $X$} \quad
1.1\\
J^{-} &   \mbox{Complementation or inversion of coordinates}
\quad 2.1\\
j^{-} & \mbox{Complementation or inversion of a coordinate} \quad 
 2.1\\
\{s,t,..,w\}^{-} & \mbox{Complementation or inversion of
coordinates} \quad    2.1\\
(F_{1},...,F_{n}) & \mbox{Transformation $F$ from
$\mathbf{Q}^{m}$ to $\mathbf{Q}^{n}$}\quad\\
 & \mbox{such that $F_{i} = p_{i}F$} \quad 2.3\\
\, [f_{1},...,f_{n}] & \mbox{[\;]-representation of a self-dual
      transformation}\quad\\
\, \langle \tau ,..,\omega \rangle  &   \mbox{A subgroup
generated by elements $\tau ,..,\omega $} \quad\\
    &    \mbox{of a group} \quad 1.2\\
\langle f \rangle & \mbox{Circular self-dual transformation
defined by} \quad\\
 & \mbox{$f: \mathbf{Q}^{n} \rightarrow  \mathbf{Q}$ such that
$p_{1}\cdot f = f$} \quad 2.4\\
 &   \mbox{Transformation defined by $f$ in circular second-order
DNNs} \quad 8.3 \\
<h^{1},h^{0}> & \mbox{Transformation defined by $h^{1}$ and
$h^{0}$ in a circular} \quad\\
 & \mbox{second-order DNN} \quad 8.3  \\
 \langle\langle f\rangle\rangle  & \mbox{Skew-circular
transformation defined by $f:
\mathbf{Q}^{n}
 \rightarrow  \mathbf{Q}$} \quad \\
 & \mbox{such that $p_{1}\cdot f = f$} \quad  2.4\\
 |f| & \mbox{Number of elements of the set $f^{-1}1$ for
a function $f$}  \quad 1.4\\
\, [x] & \mbox{The greatest integer not greater than real number
$x$} \quad 1.2\\
\, [x] & \mathrm{Orb}_{\mathbf{G}}x \mbox{ for a given group}
\mathbf{G} \quad 6.6,
7.1, 7.2, 8.2, 9.5. \\ 
\, [S] & \mathrm{Orb}_{\mathbf{G}}S \mbox{ for a given group}
\mathbf{G} \quad 6.6,
7.1, 7.2, 8.2, 9.5. \\ 
 (s_{1},s_{2},..,s_{m}) & \mbox{Cyclic permutation} \quad  1.2\\
 (x,y) & \mbox{Inner product of $x$ and $y$} \quad 4.1\\
 X^{c} &   \mbox{Complement of a subset $X$} \quad 1.1\\
 F: X \rightarrow  Y &   \mbox{$F$ is a function from $X$ to $Y$}
\quad 1.1\\
 A \rightarrow _{F} B & \mbox{$B$ is the image of $A$ under $F$}
\quad 1.1\\
 x \rightarrow  y & \mbox{$(x,y)$ is an arc of a digraph} \quad
1.1\\
 x\partial  &   \mbox{$(x,x)$ is a loop of a digraph} \quad 1.1\\
\end{array}
$$

$$
\begin{array}{ll}
 F \mid A & \mbox{Transformation $F$ restricted to $A$} \quad
1.1\\
 \backslash & \mbox{Difference of sets: $A\backslash B = \{ x
\mid  x \in  A and x \notin  B\}$}
\quad  1.1  \\
 +    &    \mbox{Sum of transformations} \quad 1.1\\
 \dot{+} & \mbox{Symmetric difference of sets: $A\dot{+}B =
(A\backslash B)\cup (B\backslash
A)$}  \quad 1.1\\
 \times    &    \mbox{Cartesian product of sets} \quad  1.1\\
    &     \mbox{Direct product of transformations} \quad  1.1\\
     &    \mbox{Direct product of groups} \quad  1.2\\
 \emptyset  &      \mbox{Empty set} \quad 1.1\\
 \circ  &   \mbox{Composition of functions or transformations}
\quad  1.1\\
 \odot  &    \mbox{Disjoint composition (direct sum) of
transformations} \quad 1.1\\
 \vee  &      \mbox{Disjunction, OR} \quad 1.4\\
 \cdot    &     \mbox{Conjunction, AND, Product} \quad 1.4\\
 \neg    &    \mbox{Complementation, Negation} \quad 1.4\\
 (=) & \mbox{Equality operation on $\mathbf{Q}^{n}$ or
$\mathbf{B}$} \quad 1.4, \\
\bar{\neg} &     \mbox{Complementation, Negation} \quad  2.1 \\
\sim _{G} &    \mbox{Equivalence relation with respect to an
acting group $G$} \quad 1.3 \\
 \sim _{f} &   \mbox{Equivalence relation} \quad 4.2\\
 \sim _{F} &   \mbox{Equivalence relation} \quad 5.2\\
 \succeq _{f} &    \mbox{Preorder} \quad 4.2\\
 \equiv  mod &   \mbox{$a \equiv  b$ mod $n$ means $a-b$ is
divisible by $n$}         \quad 1.2\\
 \%   &     \mbox{$a \% b$ is the remainder obtained by dividing
$a$ by $b$} \quad 1.2 
 \end{array}
 $$
  
\end{document}