The tensor language provides a unifying approach that simplifies notation, which leads to compact modeling of multi-way information objects in many knowledge fields, and a thought framework as well. By such a language, it is modeled a generic system that connects to environment through its boundaries.

1. GENERAL THEORY OF BOUNDARIES
1 *
Vicente Fachina
1
Project manager in Petrobras, Brazil
ABSTRACT
The tensor language provides a unifying approach that simplifies
notation, which leads to compact modeling of multi-way information
objects in many knowledge fields, and a thought framework as well.
By such a language, it is modeled a generic system that connects to
environment through its boundaries. The interactions between system and
environment are modeled by the availability and the warp tensors, which
enable the creation of a system-to-environment coupling tensor. Such an
approach sets a general framework for comparing systems to themselves,
and to one another.
KEYWORDS
Ecological economics, efficiency, input-output model, recycling
1 *
Corresponding author, vicentefachina@gmail.com

2. 2
INTRODUCTION
This paper aims at applying the tensor approach for analyzing
complex systems other than geometry or physics, such as economy and
ecology, for instance. Complex systems mean those ones with more than
one element, for which element interplays can happen. Besides geometry
or physics, the tensor approach provides a general framework for
modeling objects regardless of their complexities, should they be a living
being, an ecosystem, equipment, systems, industry, or whole economies
or ecosystems.
An object is a tensor if it is not characterized by any particular
frame of reference. For situations in which objects are characterized by
changes in coordinate systems, or whether those coordinate systems are
not possible or feasible to setup, the tensor approach does not apply, as
for instance: matrices relating coordinate system pairs (Jacobian
matrices), or intrinsic variations of a coordinate system itself, or the
simple derivative of a tensor that does not yield a tensor. Another
example of variant that is not a tensor is communication processes: if the
transmitter´s meaning does not fully match to the receiver´s one, both the
transmitter´s and the receiver´s communication systems are not related to
each other by stable translation rules (Jacobian matrices).
On the history of the tensor approach, it dates back the end of the
nineteenth century by Riemann (Riemann, 1867), and Ricci, Levi-Civita
(Ricci, et al., 1900) for solving the issue on how to describe geometric
objects independently on coordinate systems.

3. 3
Then Einstein came forth in the first two decades of the twentieth
century, and he highlighted the power of the tensor approach to the
scientific community through his General Theory of Relativity by which
space-time has become a dynamic framework.
Singh et all (Singh, et al., 2009) made a comprehensive review of
frameworks for designing sustainability indicators, which covers:
Innovation, Knowledge and Technology Indices; Development Indices;
Market-and Economy-based Indices; Eco-system-based Indices;
Composite Sustainability Performance Indices for Industries; Investment,
Ratings and Asset Management Indices; Product-based Sustainability
Index; Sustainability Indices for Cities; Environmental Indices for
Policies, Nations and Regions; Environment Indices for Industries;
Energy-based Indices; and Social and Quality of Life-based Indices. Out
of all such indicators, the Ecological Footprint, which belongs to the
Eco-system-based Indices, is the one that mostly resembles the goal of
this paper for the explicit inclusion of systems (in the form of required
resources) and environment (in the form of available resources), for
which the indicator is the ratio of the former to the latter, or the inverse:
ratios exceeding one (or below one) bring about concerns.
Burdick (Burdick, 1995) and Cichocki et al (CICHOCKI, et al.,
2009) used the tensor approach for multi-way data processing, mining,
and analyses. Nevertheless, they used the tensor algebra as an extension
to linear algebra for matrices larger than the square ones (cubic and
higher orders), and they used neither the tensor indicial notation nor the
metric tensor.

4. 4
Mathews and Tan (Mathews, et al., 2011) showed eco-efficiency
projects in China, Denmark, Korea, Japan and Australia by utilizing
flowcharts that depict waste energy and material streams from production
units that become resources to one another. They stated two performance
criteria and their relationship: (i) it must improve the eco-efficiency of
the group of companies as a whole; and (ii) it must improve the profit
position of at least one company without damaging the profit position of
the others. They did not model eco-efficiency performance metrics.
Geng et al (Geng, et al., 2012) described one-dimensional
indicators for the Chinese economy as either ratios or absolute values:
Resource output rate, Resource consumption rate, integrated resource
utilization rate, and Waste disposal and pollutant emission. They cited
some setbacks: Lack of indicators on urban/industrial symbiosis, Lack of
indicators for businesses, Lack of absolute material/energy reduction
indicators, and Lack of prevention-oriented indicators. On prevention-
oriented indicators, there is the risk of preferring solutions based on
recycling and reuse instead of solutions based on prevention and source
reduction built into the design of processes and products.
This paper applies the tensor approach for seamlessly modeling
efficiency and recycling interplays under a general framework.
SYSTEM AND ENVIRONMENT
In geometry, an object is characterized by a coordinate system.
A geometric object is defined by a set of coordinate numbers, and the
object´s identity is unchanged and independent on any changes in the
coordinate system. Tensors are arrays that describe objects independently
on frames of reference. The Appendix 1 shows a geometric example.

5. 5
In order to extrapolate the tensor approach to realms other than
geometry or physics, the definition of object has to comprise any
observable set within environment, and to be characterized by frames of
reference. A system can be any set within a larger one, for which frames
of reference can be setup. For instance, in economy, a production object
is a system within environment, which in this case can be a group of
production objects, a city or region. Such a production object can be
characterized by its flows and stocks, which can be measured by
currency units, energy or material units. The inception of the tensor
approach in realms other than geometry depends on the existence of
transformation rules among frames of reference.
Transformation rules (Jacobian matrices) are partial derivatives of
one unit as to another. For simplicity, one can assume diagonal Jacobians
only, by which the identities of units do not depend on one another. In
geometry, that means an orthonormal coordinate system.
If frames of reference can be setup, and arithmetic operations are
allowed, Inequality (1a) defines a system2
, which can be anything but
nothingness. A convention for the flow of quantities shall be adopted:
Ekk delivers to Ekj, and Ekk receives from Ejk.
Equation (1b) models a conservative system: all that Ekk delivers
shall be equal to all that Ekk receives. Equation (1b) holds true for each
time instant, thus it can comprise time derivatives as well3
. The k-index
is a summation dummy by the Einstein´s summation convention, either
as a line (1st
subscript) or a column (2nd
subscript). All Ekj tensors ought
to be invertible.
2
Systems stem from environment, and they can be themselves environment for other
lower level systems: a fractal mode, both up and downwardly.
3
In such cases, the invariant time derivative operator ( ) has to be applied lest the
tensor property be lost (Grinfeld, 2013). Further details in the Appendix 3.

6. 6
0kkE (1a)
0 k
jk
k
kj EE (1b)
All quantities must be in the same measuring unit; for other
measuring units, the Ekj tensor shall be transformed appropriately4
.
Environment is defined as a 3rd
-order tensor Zikj, with the i-index
representing environment endogenous, independent quantities, from
which the kj values stem. For modeling environment´s availability or
probability fields, the tensor ψi
kj is defined. The availability tensor ψi
kj is
a coefficient array, of which contraction with environment tensor Zikj
enables those fields to collapse into systems. Therefore, the definition
Xkj=ψi
kjZikj models environment´s availability or probability fields, and
Equation (2a) is thus created from the very identity Xkj=Xkj.
The warp tensor5
λkj stems naturally from Equation (2a), and it
models the system´s boundaries. The term “warp” refers to the recycling
level of a process: circular systems mean warp elements below one
(λkj <1).
4
A Jacobian is a matrix that transforms a tensor from a coordinate system to another.
5
There is a dual model by defining the waste tensor as   kj
j
kkjkj EW   . There is waste if
such a difference is positive (λ>1), and recycling if negative (λ<1). This definition leads
to kjkjikj
i
kj EWZ  , or j
kkj
kj
kj WE   , where the δ symbol is the unity matrix in tensor
notation. Also, one can define waste per unit system, j
kkjkj
kj
WE
E
W
  . Reference
(Nakamura, et al., 2009) presents a complete study on waste input-output analysis. An
intrinsic, bottom-up definition of the warp tensor is the pairwise dot products of the
measurement units or coordinates: jkkj   A measurement unit or coordinate can be
defined as environment differentiations as to a system: Zzz  , where z is a coordinate
or measurement unit. In non-Euclidian geometry or physics, the warp tensor is called
the metric tensor.

7. 7
In the Appendix 1, it is shown a geometric example of the warp
tensor. In the Appendix 2, it is shown a model for calculating the proper
system configuration, the warp tensor, in order to comply with
environment threshold values.
Equations (2b)6
stem from Equation (2a) for defining the Gi
kj
tensor, a system-to-environment coupling tensor: the more it approaches
the unity array the closer is the system from environment. A unity Gi
kj
tensor means system and environment are the same.
kjkjkj
kj
kjkj EEEXX  (2a)
kj
kj
iikjikj
i
kjkj
i
kj
kji
kj EGZZGEG   (2b)
Figure 1 illustrates a system, which comes to being by matching
environment´s availabilities (availability tensor) to the system´s
boundaries (warp tensor).
Boundaries may not be physical interfaces only, logical thresholds
may be so too. Therefore, for instance, Figure 1 can illustrate planned
activities versus actual ones, or business management: an element of λkj
being one indicates an actual activity fully complies with the
corresponding planned one. See Appendix 4 for further details.
6
On changes in reference, ikj
i
jkikj
i
kj
j
j
k
kkj
j
j
k
kjk ZGZGJJEJJE ´´´´´´´´  , and
i
jk
kji
kj
kjj
j
k
k
i
jk JJG ´´´´´´   , thus ikj
i
jk
kj
jk ZE ´´´´  , and   kj
kj
i
i
jkjk EE  ´´´´  , with 0i
kj ,
which allows a system to exist in another existent frame of reference. Bottom line, the
engine for changing frames of reference is j
j
k
k
kj
i
i
jk JJ ´´´´ 

8. 8
Figure 1. System and environment
Equation (3a) models the availability tensor by the exergy
approach, which consists on applying a complete Legendre
transformation7
(Callen, 1985) over environment´s quantities (zi), and by
assuming a Cartesian environment (independent environment´s
endogenous variables, a flat environment).
The second terms on the right make up environment irreversibility.
Equation (3a) holds true for each time instant, thus it can comprise time
derivatives as well.
Equation (3b)8
models the availability tensor by an economic
approach, which is useful for high aggregated economic systems.
7
A complete Legendre transformation substitutes all the function´s first derivatives for
the function itself. A complete Legendre transformation for a scalar function
is , where ψ is an availability factor (exergy efficiency if Z is an
energy field), Z is environment, and the i-index is for environment´s endogenous
variables (z). Also, there are references on exergy analyses for all equipment (Pavelka,
et al., 2015). A ψkj element can be over unity if the summation over environment´s
irreversibility turns out to be negative.
8
Equation (3b) is designed for resembling Equation (3a): Present Value (PV) is
equivalent to Z, and I is equivalent to irreversibility. The difference PV-I is the
definition of Net Present Value (NPV). If ψ is a diagonal tensor, then .

9. 9
ikjz
ii
ikjikj
i
kj ZdzZZ  (3a)
i
ikj
i
ikjikj
i
kj IPVPV  (3b)
The i-index is a summation dummy by the Einstein´s summation
convention.
VARIATION ANALYSES
The goal is to elaborate a useful invariant out of Equation (2a) in
order to evaluate systems, either against themselves with time or against
other systems.
Equation (4) models a structure tensor that yields the relative
weight of a system´s element as to the whole system. The denominator
means the summation of all elements, the upper indexes being
summation dummies9
.
Equation (5a) defines a working variable from Equation (2a) and
Equation (4). Equation (5b) stems from the product λkjSkj. Equation (5c)
takes the determinant of Equation (5b). For simple notation, the capital
letters in Equation (5c) indicate determinants.
kjjk
kj
kj
XEA  (5a)
k
j
kj
kjkj AS   (5b)
1SA (5c)
9
jk
kj
kj
kj
E
E
S  (4)

10. 10
Equation (6) models the covariant derivative (which takes into
account variations in the coordinate system as well) of Ekj as to any v
independent variable, and endogenous to the system, but time10
. The m
index is a dummy for the operation of the Christoffel´s symbol. Also, to
note the covariant derivative reduces to the partial derivative if the
coordinate system is constant.
mk
m
vjmj
m
vkkjvkjv EEEE  (6)
Equation (5c) is a scalar one for it involves determinants only, thus
Equation (6) reduces to partial derivatives, which yield Equation (7a).
In case of an economic system, for instance, Equation (7a) relates
three scalar quantities to one another: (i) the activity effect (A); (ii) the
efficiency effect (λ); (iii) the structure effect (S).
Equation (7b) is a special case of Equation (6), for which Ekj is a
constant but an invariant. The variations of Ekj as to v are balanced by
intrinsic variations of the coordinate system, thus to yield a constant Ekj.
0 ASASSA  (7a)
mk
m
vjmj
m
vkkjv EEE  (7b)
EXAMPLES
Two examples are shown: a 3x3 accounting system, and a 4x4
production system. It is assumed such systems make up tensors11
, for
they do not depend on measuring units for their very existences.
10
In economy, variable frames of reference can be inflation, interest rates, or exchange
rates; in geometry or physics, variable coordinate systems are curvilinear ones, as on
warped surfaces or space-time with energy.
11
A way of proving a variant is a tensor is to use the Jacobian for any two coordinate
systems: be for instance the gradient of an E scalar field in the coordinate system,
; in the coordinate system, it is ; since ,

11. 11
On the first example, Table 1a shows currency values on two
periods for an accounting system, with environment to be either
investment or financing realms. In order to follow accounting rules, both
system and environment do follow Equation (1b). The diagonal values
(boxed) are the currency assets, and the off-diagonal ones are their
interplays. By following Equation (1b): E11+E21+E31=E11+E12+E13,
E12+E22+E32=E21+E22+E23, and E13+E23+E33=E31+E32+E33. The same
applies to X. In Table 1a, E33, X33 have changed (in bold). In such cases,
system and environment are said to be “closed”.
Table 1a. System and environment
T0 T1
E - System
150 70 30
70 200 50
30 50 300
E - System
150 70 30
70 200 50
30 50 350
X - Availabilities
200 250 100
200 400 350
150 300 600
X - Availabilities
200 250 100
200 400 350
150 300 550
Table 1b shows the derivatives by Equation (7a) on both T0 and T1,
and also the mean values. The conclusions are: (a) In absolute values,
ΔS∂S<ΔA∂A<Δλ∂λ, which means the efficiency effect dominates,
followed by the activity effect; (b) Some residue may show up as the
derivatives in Equation (7a) considers continuous function.
, which proves is a tensor. Therefore, the gradient of any
scalar field is a tensor. Coordinates do change, not the gradient field itself.

12. 12
Table 1b. ∆t comparisons
ΔA∂A Δλ∂λ ΔS∂S Residue
On T0 0.3692 -0.2748 0.0071 0.1015
On T1 0.2696 -0.3790 0.0071 0.1022
Mean 0.3194 -0.3269 0.0071 0.0004
On the second example, Table 2 shows a 4x4 production system
that comprises three independent energy facilities, and a production unit.
The net deliveries from the three energy facilities are: 700 MWh/ha
windpower, 500 MWh/ha solar PV power, and 4,800 MWh/ha
geothermal power. The 4th
element is the production unit, which utilizes
the energy from those three independent energy facilities with 5%
transmission loss, thus it consumes 5,700 MWh/ha actually. The three
gross energy resources are: 1,556 MWh/ha wind, 1,852 MWh/ha solar
PV, and 13,713 MWh/ha geothermal. Nevertheless, there are 59% wind
power availability, 50% solar PV power availability, and 60%
geothermal power availability. The reasons for these availabilities are:
(a) For wind power, there is the 59% upper limit12
for maximum
energy efficiency;
(b) For solar PV power, there is the 50% efficiency roadmap13
;
(c) For geothermal power, there is the 60% Carnot efficiency for a
782K hot source and a 313K heat sink, for instance.
Therefore, Table 3 comprises environment´s net deliveries:
918 MWh/ha wind, 926 MWh/ha solar PV, 8,228 MWh/ha geothermal,
and 10,072 MWh/ha total available.
12
Betz´s law states the maximum energy that can be extracted by a wind turbine in open
flow is 59.3% out of the incoming wind kinetic energy. In this case, 0.593 is the exergy
factor (ψ) by Equation (3a).
13
Based on the Fraunhofer Institute for Solar Energy Systems, 44.7% efficiency is the
world-record energy efficiency achieved by concentrated PV array (CPV) in 2013. In
this case, the 0.50 efficiency roadmap is the exergy factor (ψ) by Equation (3a).

13. 13
From Tables 2, 3, Equation (2a) is solved for λkj and then taken the
determinant, which yields λ=7.3566. That means 13.6% actual efficiency.
Table 2. Production system Ekj; units in MWh/ha
Wind
power
PV solar
power
Geothermal
power
Production
unit
700 0 0 665
0 500 0 475
0 0 4,800 4,560
0 0 0 5,700
Table 3. Environment´s availability Xjk; units in MWh/ha
Available
wind
Available
PV
Available
Geothermal
Total
availability
918 0 0 0
0 926 0 0
0 0 8,228 0
0 0 0 10,072
In case of technological upgrades, if the system is supposed to
achieve 85% energy outputs out of environment´s net availabilities, and
2% transmission loss, Equation (2a) yields λ=1.9548, which can be set as
an overall efficiency target. That means 51% efficiency target.
Table 4 stems from Tables 2,3 for showing how Equation (1b)
morphs into the energy conservation law by creating a closed system,
which comprises both the original system and environment. Such a
closed system is brought about by making λkj=δk
j, which yields Xkj=Ekj.

14. 14
The 5th
column stores losses or wastes, and the 5th
line stores
environment´s net deliveries. Cell (5,5) stores total losses or wastes14
.
Table 4. Closed system; units in MWh/ha
Available
wind
Available
PV
Available
Geothermal
Production
unit
Wastes, losses to
environment
918 0 0 700 218
0 926 0 500 426
0 0 8,228 4,800 3,428
0 0 0 5,700 300
918 926 8,228 -5,700 4,372
ENTROPY
This section presents a quantitative concept for assessing system´s
scenarios. Such a concept is entropy, for it yields positive values, and it
has an absolute minimum: zero, which means no missing information.
Entropy measures missing information in a finite scheme
(Khinchin, 1957). The more entropy the more missing information: zero
entropy means there is no uncertainty in a system, no missing
information. A finite scheme is defined as a finite system with a finite
number of states, each of which with a certain probability to occur. Also,
a finite scheme assumes one and only one state at each time (this
assumption does not hold in the quantum realm). Equation (8) defines
entropy in a finite system.
14
Wastes can be recuperated by technological, behavioral upgrades; losses cannot.

15. 15
     
 
  0
1
log





mkj
m
mkj
m
mkjmkjkjkj
Ep
Ep
EpEpEH (8)
By Equation (2b), a system stems from environment´s
availabilities. Actually, system and environment are mutually dependent
on each other.
Equation (9) stems from Equation (8) and Equation (2b): given a
environment (Zikj), Equation (10) calculates a system´s entropy from the
coupling tensor and environment.
Equation (10) is a conditional probability dependence of the
coupling tensor on environment.
Equation (11) calculates entropy of an entire system. Equation (12)
draws from Equation (1b), and it presents an alternate way for calculating
entropy of an entire system by defining a conservative system as having
no missing information, or zero entropy; nevertheless, it is a different
variant as to Equation (11). Probability depends on frames of reference,
thus Equations (11), (12) can yield different values for different frames
of reference.
     Z
i
kjkjikjkjikj
i
kjkj GHZHZGH  (9)
   l
i
kj
l
kjlZ
i
kjkj GHpGH  (10)

16. 16
Therefore, entropy is not a tensor15
.
   kj
jk
kj EHEH  (11)
  k
k
k
k
k
jk
k
kj
E
EE
EH

*
(12)
Equation (13) states that for any system and environment, overall
entropy increases (system entropy plus environment entropy). For
energy, this resembles the second law of thermodynamics. Refer to
Appendix 3 for invariant time derivative.
By Equation (13), decreasing entropy in a system can occur if does
occur greater increasing entropy in environment, or decreasing entropy in
environment can occur if does occur greater increasing entropy in a
system.
Now, a flat environment is defined as containing non-entangled
energy quantities, thus with maximum entropy. By random quantum
fluctuations, such energy quantities may eventually get entangled, and a
system is created, thus entropy is lowered. With enough time, lots of
types of structure may emerge, life included, due to lower entropy.
15
Ekj is a tensor, thus j
j
k
kkjjk JJEE ´´´´  , whose entropy is:
           
m
m
j
j
k
kkjm
j
j
k
kmkjjkjk JJpEpJJpEpEH ´´´´´´´´ loglog
Therefore, entropy is a variant but a tensor. If and only if there is no uncertainty in the
transformation rules among frames of reference,   1´´ j
j
k
k JJp , which implies
   kjkjjkjk EHEH ´´´´´  . In such a particular case, entropy is a tensor. Entropy measures
missing information, thus a flat environment has maximum entropy. Therefore, one can
define information as any energy change with lower entropy than environment.
Asymmetries in energy fields create information, so energy comes first, then followed
by information.
      00  i
kjkjikjkjkjkj GHZHEH  (13)

17. 17
CONCLUSIONS
For comparing complex, distinct, heterogeneous systems to one
another, it is necessary to create a general framework, by which all
systems can be compared to one another, besides to themselves. In order
to create such a general framework, the language of the tensor approach
is borrowed. In this way, measuring units become equivalent to
coordinate systems. Also, systems are defined as to environment, within
which they are embedded.
A system is defined as anything but nothingness by a 2nd
-order
tensor array, of which the diagonal values are different from zero, and the
off-diagonal ones represent interplays. The relationship between system
and environment emerges from an identity involving environment´s
availabilities, by which one models system´s boundaries through the
warp tensor, and the coupling tensor between system and environment.
Also, an invariant scalar equation is developed for comparing systems,
either to themselves or to one another.
For assessing system´s scenarios, the concept of entropy is applied.
Entropy turns out to be a tensor in a special case when there is no
uncertainty in the transformation rules among frames of reference;
otherwise, entropy is not a tensor, thus it depends on frames of reference.
ACKNOWLEDGEMENTS
I thank Petrobras for its R&D on energy efficiency and renewable
energy.

18. 18
APPENDIX 1 – A GEOMETRIC EXAMPLE
The purpose of this appendix is to use a geometric example for
illustrating an intrinsic, bottom-up building of the warp tensor. For
clarifying the concept of the warp tensor embedded in Equation (2a),
Figure 2 shows a sphere embedded within a z1, z2, z3 Cartesian
environment.
Equation (14a) defines a measuring unit by partial differentiation of
environment as to a coordinate. Equation (14b) is the partial
differentiation of environment as to the spherical coordinates r, θ, φ,
which is the algebraic form of the Cartesian-to-spherical Jacobian.
Equation (14c) defines the warp tensor by the pairwise dot products of
measurement units, or coordinates.
Zkk

 (14a)
  321 cossinsincossin,, zrzrzrrZ

  (14b)
jkkj 

 (14c)
Equation (15) is the warp tensor of a sphere, which is a varying
second-order, 3x3 diagonal matrix. For instance, if a system is a hyper
plane, the warp tensor shall have constant elements.
Equation (16) defines length or distance as a primary concept16
. For
other systems, such a metric depends on defining measuring units for
building a proper warp tensor.
16
For a sphere, 2222222
sin  drdrdrdL  ; for a spherical surface of radius R,
222
sin  ddRdL  .

19. 19













22
2
sin00
00
001
r
rkj (15)
ZZdL jkkj  2 (16)
Figure 2. A sphere within a Cartesian environment
Z
θ
φ
λr
z3
z1
z2
λθ
λφ

20. 20
APPENDIX 2 – THE FOSSIL INDUSTRY
The fossil industry comprises coal, oil and gas, of which
combustion processes discard carbon dioxide and other harmful elements
that drive global warming in the 21st
century. The fossil industry supply
about 81% of the total energy consumed in the planet as of 2015. Even if
all the climate change pledges are fully implemented by 2040, the fossil
industry shall still represent about 74% of the total energy consumed by
that time (2016 World Energy Outlook, iea.org).
Recycling the fossil industry can be defined as utilizing the carbon
dioxide17
from combustion processes back into the economy instead of
discarding it to biosphere. About half of the carbon dioxide released from
combustion processes remains in the atmosphere, and it is not absorbed
by vegetation and the oceans (Ramsayer, 2015).
For reducing the warp tensor to values below one in Equation (2a),
besides lower activity effect and renewable energy (structure effect),
economic utilization of carbon dioxide shall reduce the need for fossil
fuels. Routes for reducing fossil fuel use by recycling carbon dioxide can
be isobutyraldehyde (Atsumi, et al., 2009), methanol (Cifre, et al., 2007),
and carbon monoxide for liquid fuels (Sun, 2016).
Equation (17) calculates the waste tensor for keeping environment
variables within thresholds without reducing the activity effect.
Equation (18) calculates the warp tensor from the waste tensor.
Such warp factors set recycling targets by Equation (2a).
17
Recycling CO2 can reduce fossil fuel consumption, and also make it emission-neutral,
thus making a transition from CCS-Carbon Capture and Storage towards CCU-Carbon
Capture and Utilization.

21. 21
 thresholdskjkj WW  (17)
j
kkj
kj
kj WE   (18)
Due to its much higher energy density than solar, wind, or biomass,
fossil fuels shall not be displaced by renewable energy alone. Natural gas
shall be a bridge in the 21st
century as a relatively cleaner energy carrier.
Either gaseous or liquefied, natural gas can be used in proper fuel cell
systems18
, from which CO2 emission shall be quite less than by
combustion processes. In the other side of that energy transition bridge,
the fossil industry may live on as supplier of petrochemical raw
materials, and much less as a fossil energy supplier.
The activity effect may be negative because of projected nine
billion people by 2050 in the planet. This can be counteracted by the
efficiency effect and the structure effect for reducing energy demands
from fossil energy carriers.
On the structure effect, there can happen disruptive energy
technologies: thorium-based nuclear fission energy, and nuclear fusion
energy.
18
Besides hydrogen and methane, methanol, ethanol and ammonia as well can deliver
electricity by utilizing them in proper fuel cells (Qi, 2014).

22. 22
APPENDIX 3 – INVARIANT TIME DERIVATIVE
In Equation (6), time is not allowed in the derivatives for time is
not endogenous to classical systems; space-time is an environment
variable only. In such a case, the invariant time derivative shall be
applied for keeping the tensor property; otherwise, partial time
derivatives can work only if the frame of reference is constant. Further
details on the invariant time derivative operator can be found in Grinfeld
(Grinfeld, 2013).
With the invariant time derivative as a background tool, it is worth
analyzing a stability criterion for a system. A system is stable if its values
along time do not get beyond a finite boundary. Concave functions can
model such a stability criterion.
Equation (19) stems from Equation (2b) by applying the invariant
time derivative as function of the system-to- environment coupling tensor
and environment itself, and by assuming stable environment.
Inequality (20) follows on the stability criterion. Therefore, for a
system to be stable, the system-to-environment coupling tensor must
follow the stability criterion. By developing Inequality (20), one reaches
at Inequality (21), which models the interplay condition between the
availability and the warp tensors for keeping a system stable.
i
kjikjkj GZE 22
  (19)
02
 i
kjG (20)
  kji
kj
i
kj
kjkji
kj    22
2
1
(21)

23. 23
Equation (22) models a system with full synergy as to environment
by applying the invariant time derivative to a system-to-environment
coupling tensor being the unity array. This can be thought of as wave
equations.
0 kj
i
i
kjkj
kj
  (22)
Given environment availability dynamic functions, Equation (22)
can be solved for the warp tensor in order to setup a system with full
synergy as to environment.

24. 24
APPENDIX 4 – BUSINESS MANAGEMENT
This appendix illustrates the application of Equation (2a) on
business management, which comprises asset, process, project, program,
or portfolio management. Basically, management activities deal with
comparing planned activities to actual ones, in order to close the gaps
among them for scope, quality, and time.
Equation (23) stems from Equation (2a) by solving for λkj, and it
models the comparison between actual activity array, Ekj, and planned
activity array, Xkj. If all the actual activity measurements match to the
planned activity targets, λkj resembles the unity matrix, for which the
determinant is one.
Figure 3 illustrates a control chart that shows how much the actual
activity measurements (the dotted curve) match to the planned activity
targets on each period. If an actual activity measurement is out of range
(the upper bar in the graphic), management should either set up a recoup
plan or re-plan for the next periods, in order to keep the activities on
track, below the tolerance threshold.
kj
kjEX (23)
Figure 3. Actual-to-plan assessment

Time
0
1

25. 25
By building on the Equation (3b), performance metrics can be
derived for investment projects in general. Equation (24) stems from
Equation (2a) and Equation (3b). In this case, the warp tensor contains
factors on time and interest ratio that translate future values into present
ones; otherwise, there would be a flat scenario, which means a unity
warp tensor, with no interest ratios or future time.
kjkjkjkj EIPV  (24)
Equation (25) defines the Return-Over-Investment metric, ROI.
The ROI tensor contains ratios of net present values to investment values.
Equation (26) stems from Eq. (24) and Equation (25). The ξ-tensor
contains ratios of business values to investment ones. The larger the ROI
the better.
Equation (27) defines the Levelized Cost Of Product, LCOP. The
LCOP tensor contains ratios of present cost values to product yields
along a defined period. The smaller the LCOP the better.
Now, by dropping the tensor indexes, from Equations (26), (27)
one can derive a relationship between ROI and LCOP. The simplifying
hypotheses are constant currency flows along time, and non-zero
investment. By defining I0 as a reference investment, Equation (28)
defines (ROI)0 by considering an opportunity E0 value. By defining Y0 as
a reference yield, Equation (29) is another form of Equation (27).
  kj
kjkjkj IIPVROI  (25)
kjkjkj
kj
kjkj ROIIE   (26)
 kjkj
j
k
kj
kj
kj
kj
kj
t
IY
t
PVY
 



LCOP (27)

26. 26
Equation (30) relates LCOP to (ROI)0. At last, by defining (ROI)0
cost
,
Equation (31) is another form of Equation (30).
Equation (31) is displayed as a linear plot, with two notable points:
(i) LCOP (ROI=0)=μν; (ii) LCOP (ξ=-1/λ)=0, or LCOP (ROIcost
=ν)=0.
0I
I

  








 0
0ROI (28)


 00

I
E
I
E
0Y
Y

   1LCOP (29)
0
0
0
Y
I
c 
t
c




 0
  0ROILCOP   (30)


 0
1
      tt
ROIROIROI
cos
00
0cos
0 





 


   t
ROILCOP
cos
0  (31)

27. 27
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