4.2.3. Biodiesel

Algal growth in water bodies is an indication of water pollution. However, nutrient rich waste water can be utilized for the production of blue-green algae that accumulate the lipids in it and that in turn can be utilized for the production of biodiesel trough transesterification [69,70]. Algal biomass thus obtained can also be utilized as animal feed and also spread out as fertilizer. Certain species of algae, grown in wastewater is capable of produce oil up to 80% as its storage product and produce more oil, around 23 times more than the best oil-seed plant. In terms of oil produced per unit area, algal productivity exceeds palm oil by 10 bend, and jatropha, canola, and sunflower crops by more than 30 bend [20]. Oil transesterification to biodiesel, algal biomass, and glycerol can be utilized to produce energy products, and glycerol may be used as fuel for burning directly or can be converted to hydrogen and bioethanol by fermentation [71]. The viability of algal biomass for the production of biodiesel can be improved by instant wastewater treatment, use as animal feed, and the production of secondary energy products. Microalgae were also used in phytoremediation of wastewater sources [51].


# **The Impact of the COVID-19 Emergency on the Quality of Life of the General Population**

Edited by Michele Roccella

Printed Edition of the Special Issue Published in *Journal of Clinical Medicine*

www.mdpi.com/journal/jcm


*Article* **Proof-of-Concept of a Quasi-2D Water-Quality Modelling Approach to Simulate Transverse Mixing in Rivers**

**Pouya Sabokruhie 1 , Eric Akomeah 2 , Tammy Rosner 3 and Karl-Erich Lindenschmidt 1,\***

	-<EMAIL_ADDRESS>
**Abstract:** A quasi-two-dimensional (quasi-2D) modelling approach is introduced to mimic transverse mixing of an inflow into a river from one of its banks, either an industrial outfall or a tributary. The concentrations of determinands in the inflow vary greatly from those in the river, leading to very long mixing lengths in the river downstream of the inflow location. Ideally, a two-dimensional (2D) model would be used on a small scale to capture the mixing of the two flow streams. However, for large-scale applications of several hundreds of kilometres of river length, such an approach demands too many computational resources and too much computational time, especially if the application will at some point require ensemble input from climate-change scenario data. However, a one-dimensional (1D) model with variables varying in the longitudinal flow direction but averaged across the cross-sections is too simple of an approach to capture the lateral mixing between different flow streams within the river. Hence, a quasi-2D method is proposed in which a simplified 1D solver is still applied but the discretisation of the model setup can be carried out in such a way as to enable a 2D representation of the model domain. The quasi-2D model setup also allows secondary channels and side lakes in floodplains to be incorporated into the discretisation. To show proof-of-concept, the approach has been tested on a stretch of the lower Athabasca River in Canada flowing through the oil sands region between Fort McMurray and Fort MacKay. A dye tracer and suspended sediments are the constituents modelled in this test case.

**Keywords:** lower Athabasca River; oil sands region; quasi-2D modelling; water-quality analysis simulation program (WASP); water-quality modelling


Cell Death and Disease

*Edited by Ke Xu*

## Cell Death and Disease *Edited by Ke Xu*

Published in London, United Kingdom

Cell Death and Disease http://dx.doi.org/10.5772/intechopen.104360 Edited by Ke Xu

### Contributors

Leisheng Zhang, Baoquan Song, Yanbo Shi, Junyong Zhang, Kaitao Lu, Sunfeng Pan, Hanqiang Shi, Lie Xiong, Shuqin Du, Ting Li, Qirong Wen, Qingfa Zeng, Chayan Munshi, Shelley Bhattacharya, Vincenzo Carafa, Lucia Altucci, Antonio Beato, Gregorio Favale, Giulia Verrilli, Laura Della Torre, Vincenza Capone, Daniela Carannante, Silva Samson Abrahamyan, Karina Galoian

### © The Editor(s) and the Author(s) 2023

The rights of the editor(s) and the author(s) have been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights to the book as a whole are reserved by INTECHOPEN LIMITED. The book as a whole (compilation) cannot be reproduced, distributed or used for commercial or non-commercial purposes without INTECHOPEN LIMITED's written permission. Enquiries concerning the use of the book should be directed to INTECHOPEN LIMITED rights and permissions department (permissions@intechopen.com).

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#### Notice

Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book.

First published in London, United Kingdom, 2023 by IntechOpen IntechOpen is the global imprint of INTECHOPEN LIMITED, registered in England and Wales, registration number: 11086078, 5 Princes Gate Court, London, SW7 2QJ, United Kingdom

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library

Additional hard and PDF copies can be obtained from<EMAIL_ADDRESS>
Cell Death and Disease Edited by Ke Xu p. cm.

This title is part of the Biochemistry Book Series, Volume 39 Topic: Cell and Molecular Biology Series Editor: Miroslav Blumenberg Topic Editor: Rosa María Martínez-Espinosa

Print ISBN 978-1-83768-244-7 Online ISBN 978-1-83768-245-4 eBook (PDF) ISBN 978-1-83768-246-1 ISSN 2632-0983


*4.1.1 The comparison of a P300m peak of MEG response with nonsmelling and smelling "Zuko" incense*

Though the typical examples are the results of P300m MEG responses of only two subjects, we can find out the results as follows:

From these typical results of P300m responses, it can be considered that a possibility of the improving cognitive ability of P300 peak was shown by smelling "Zuko" incense using "auditory oddball paradigm."


3. Exact first-order finite differences derivative for the exponential function

Let us consider the exact backward and forward finite differences derivatives of ev x, at xj, given by

$$(D\_\vartheta e^{\boldsymbol{v}\cdot\boldsymbol{x}})\_{\circ} \coloneqq \frac{e^{\boldsymbol{v}\cdot\boldsymbol{x}\_{\circ}} - e^{\boldsymbol{v}\cdot\boldsymbol{x}\_{\circ -1}}}{\chi\_2(\boldsymbol{v}, \boldsymbol{j} - 1)} = \boldsymbol{v}\,e^{\boldsymbol{v}\cdot\boldsymbol{x}\_{\circ}} \text{ and } \left(D\_{\boldsymbol{f}}e^{\boldsymbol{v}\cdot\boldsymbol{x}}\right)\_{\circ} \coloneqq \frac{e^{\boldsymbol{v}\cdot\boldsymbol{x}\_{\circ} + 1} - e^{\boldsymbol{v}\cdot\boldsymbol{x}\_{\circ}}}{\chi\_1(\boldsymbol{v}, \boldsymbol{j})} = \boldsymbol{v}\,e^{\boldsymbol{v}\cdot\boldsymbol{x}\_{\circ}},\tag{6}$$

where v∈ C can be a pure real or pure imaginary constant, and the spacing functions χ1ð Þ v; j and χ2ð Þ v; j are defined as

$$\chi\_1(\upsilon, j) \coloneqq \frac{e^{\upsilon \cdot \Lambda\_{\dot{\gamma}}} - 1}{\upsilon} \cong \Lambda\_{\dot{\gamma}} + \frac{\upsilon}{2} \Lambda\_{\dot{\gamma}}^2 + O\left(\Delta\_{\dot{\gamma}}^3\right). \tag{7}$$

and

2. Exact first-order finite differences derivatives of functions

2.1. Backward and forward finite differences derivatives

a partition P ¼ f g x1; ; x2; ⋯; xN of N non-uniformly spaced points xj

� �≔ g xj

spacing function Δj, is obtained by solving the above equality for χ2ð Þj ,

� � � g xj � Δj�1 � �

g0 xj

Df g � � xj

χ1ð Þj ≔ g xj þ Δj

finite differences derivative, as can be seen in Eqs. (2) and (4).

This is an expression which is valid for points xj different from the zeroes of g0

� �≔ g xj þ Δj

� �

� � ¼ 1

� � � g xj

g0 xj

ð Þ Dbg xj

χ2ð Þ j � 1 ≔ g xj

A definition for forward finite differences at xj is

valid for points different from the zeroes of g0

differences derivatives.

164 Numerical Simulations in Engineering and Science

requirement that

where

In this section, we intend to introduce a finite differences derivative, which has the same eigenfunction as for the continuous variable case. We start with results valid for any function, but we will concentrate, later in the chapter, on the exponential function because that function is used to perform translations along several directions in the quantum realm. The resulting derivative operator will depend on the point at which it is evaluated as well as on the partition of the interval and on the function of interest. This is the trade-off for having exact finite

An exact, backward, finite differences derivative of an absolutely continuous function g xð Þ (this class of functions is the domain of the momentum operator in Quantum Mechanics), on

> � � � g xj � Δj�1 � � χ2ð Þ j � 1 ¼ g0 xj

where Δj ¼ xjþ1 � xj and the spacing function χ2ð Þj , which is a replacement for the usual

g0 xj � �X∞ k¼1

� � � g xj

g0 xj � �X∞ k¼1

ð Þx . These definitions coincide with the usual finite differences derivative when the function to which they act on is the linear function g xð Þ¼ a0 þ a1x, a0, a1 ∈ C. An exact finite differences derivative of other functions need of more terms than the one found in the usual definition of a

� � χ1ð Þj ¼ g0 xj

> 1 k! gð Þk xj � � Δj � �k

� � ¼ 1

� �N

ð Þ �1 k�1

k! gð Þk xj

� �Δk

1 , is defined through the

j�1: (2)

ð Þx .

� �, (3)

, (4)

� �, (1)

$$\chi\_2(v,j) := \frac{1 - e^{-v \cdot \Delta\_{\dot{j}}}}{v} \cong \Delta\_{\dot{j}} - \frac{v}{2}\Delta\_{\dot{j}}^2 + O\left(\Delta\_{\dot{j}}^3\right). \tag{8}$$

Xm j¼n

j ¼ ghxjþ1

v hx ð Þ jþ1 �h x ð Þ ð Þj �1

� � ¼ 1 � e

where 1 ≤ n < m < N, and

where 1 < n < m ≤ N.

Df ghx ð Þ ð Þ � �

where

χ1 v; Δh,j � � ¼ e

where χ2 v; Δh,j�1

functions is

χ1ð Þ v; j v ev xj ¼ Xm

Xm j¼n

Chain rule. The finite differences versions of the chain rule are

� � � � � ghxj

Df g � �

v and Δh,j ¼ h xjþ1

�v hx ð Þ ð Þj �h xð Þ j�1 � �=v,

ð Þ Dbg j ≔v

j ¼ gjþ1hjþ1 � gj

¼ gjþ1 Df h � �

where 1 ≤ j < N. Also, for the backwards derivative, we have

and h xð Þ is any absolutely continuous complex function on ½ � a; b .

Df gh � �

j ≔v

ð Þ Dbghx ð Þ ð Þ j ¼ ð Þ Dbg j

ghxj

1 � e

hj χ1ð Þ v; j ¼ gjþ1

j þ hj Df g � �

The derivative of a product of functions.  Hereafter, the exact finite differences derivatives that we will consider are

$$(D\_b g)\_j \coloneqq \frac{g\_j - g\_{j-1}}{\chi\_2(v, j-1)} \text{ and } \left(D\_f g\right)\_j \coloneqq \frac{g\_{j+1} - g\_j}{\chi\_1(v, j)},\tag{9}$$

with χ1ð Þ v; j and χ2ð Þ v; j given in Eqs. (7) and (8), and some properties of these definitions follow. There is a plot of χ1ð Þ v; j in Figure 2. The spacing function χ1ð Þ v; j is defined for finite values of v and Δj.

The summation of a derivative. As is the case for continuous systems, the summation is the inverse operation to the derivative,

$$\sum\_{j=n}^{m} \chi\_1(v,j) \left( D\_f \mathbf{g} \right)\_j = \sum\_{j=n}^{m} \left( \mathbf{g}\_{j+1} - \mathbf{g}\_j \right) = \mathbf{g}\_{m+1} - \mathbf{g}\_{n'} \tag{10}$$

where 1 ≤ n < m < N, and

$$\sum\_{j=n}^{m} \chi\_2(v, j-1)(D\_b \mathbf{g})\_j = \mathbf{g}\_m - \mathbf{g}\_{n-1'} \tag{11}$$

where 1 < n < m ≤ N.

The exponential function is also an eigenfunction of the summation operation. The usual integral of the exponential function also has its equivalent expression in exact finite differences terms

Figure 2. Three-dimensional plot of χ1ð Þ v; j for the exponential function ev x.

Exact Finite Differences for Quantum Mechanics http://dx.doi.org/10.5772/intechopen.71956 167

$$\sum\_{j=n}^{m} \chi\_1(v,j)v \, e^{v \cdot x\_j} = \sum\_{j=n}^{m} \chi\_1(v,j) \left( D\_f e^{v \cdot x} \right)\_j = e^{v \cdot x\_{n+1}} - e^{v \cdot x\_n} \,. \tag{12}$$

where 1 ≤ n < m < N, and

$$\sum\_{j=n}^{m} \chi\_2(v, j-1)v \; e^{v \; \ge \; v \; \ge} = e^{v \; \ge n} - e^{v \; \ge n-1},\tag{13}$$

where 1 < n < m ≤ N.

Chain rule. The finite differences versions of the chain rule are

$$\begin{split} \left( D\_{f} \mathbf{g} (h(\mathbf{x})) \right)\_{j} &= \frac{\mathbf{g} \left( h(\mathbf{x}\_{j+1}) \right) - \mathbf{g} \left( h(\mathbf{x}\_{j}) \right)}{\chi\_{1}(\mathbf{v}, j)} = \upsilon \frac{\mathbf{g} \left( h(\mathbf{x}\_{j+1}) \right) - \mathbf{g} \left( h(\mathbf{x}\_{j}) \right)}{\mathbf{e}^{\upsilon} \left( h(\mathbf{x}\_{j+1}) - h(\mathbf{x}\_{j}) \right)} \frac{\mathbf{e}^{\upsilon} \left( h(\mathbf{x}\_{j+1}) - h(\mathbf{x}\_{j}) \right)}{\upsilon \chi\_{1}(\mathbf{v}, j)} - 1}{\upsilon \chi\_{1}(\mathbf{v}, j)} \\ &= \left( D\_{f} \mathbf{g} \right)\_{j} \frac{\chi\_{1}(\mathbf{v}, \Delta\_{h,j})}{\chi\_{1}(\mathbf{v}, j)} \end{split} \tag{14}$$

where

χ2ð Þ v; j ≔ 1 � e�v Δj

ð Þ Dbg j ≔ gj � gj�1

χ1ð Þ v; j Df g � �

Xm j¼n

Figure 2. Three-dimensional plot of χ1ð Þ v; j for the exponential function ev x.

that we will consider are

166 Numerical Simulations in Engineering and Science

values of v and Δj.

inverse operation to the derivative,

where 1 ≤ n < m < N, and

where 1 < n < m ≤ N.

Xm j¼n

v

ffi Δj � v 2 Δ2

Note that we recover the usual definitions of a finite differences derivative in the limit Δj ! 0 (N ! ∞) in which case χ1ð Þ¼ v; j χ2ð Þ! v; j Δj. Hereafter, the exact finite differences derivatives

χ2ð Þ v; j � 1 and Df g � �

with χ1ð Þ v; j and χ2ð Þ v; j given in Eqs. (7) and (8), and some properties of these definitions follow. There is a plot of χ1ð Þ v; j in Figure 2. The spacing function χ1ð Þ v; j is defined for finite

The summation of a derivative. As is the case for continuous systems, the summation is the

The exponential function is also an eigenfunction of the summation operation. The usual integral of the exponential function also has its equivalent expression in exact finite differences terms

gjþ1 � gj � �

j ¼ Xm j¼n

j þ O Δ3 j � �

j ≔ gjþ1 � gj

χ2ð Þ v; j � 1 ð Þ Dbg j ¼ gm � gn�1, (11)

: (8)

χ1ð Þ v; j , (9)

¼ gmþ1 � gn, (10)

$$\mathbf{g}\left(\mathbf{D}\_{\mathbf{f}}\mathbf{g}\right)\_{\mathbf{j}} := \upsilon \frac{\mathbf{g}\left(\mathbf{h}\left(\mathbf{x}\_{\mathbf{j}+1}\right)\right) - \mathbf{g}\left(\mathbf{h}\left(\mathbf{x}\_{\mathbf{j}}\right)\right)}{\mathbf{e}^{\upsilon}\left(\mathbf{h}\left(\mathbf{x}\_{\mathbf{j}+1}\right) - \mathbf{h}\left(\mathbf{x}\_{\mathbf{j}}\right)\right)},\tag{15}$$

$$\chi\_1(\mathbf{v}, \Delta\_{b,j}) = \frac{\varepsilon^{\frac{r}{k}\left(\mathbf{z}\_{j+1}\right) - \mathbf{z}\left(\mathbf{z}\_j\right)} - 1}{v} \text{and } \Delta\_{b,j} = h\left(\mathbf{x}\_{j+1}\right) - h\left(\mathbf{x}\_j\right), \text{ and}$$

$$(\mathbf{D}\_b \mathbf{g}(h(\mathbf{x})))\_j = (\mathbf{D}\_b \mathbf{g})\_j \frac{\chi\_2\left(\mathbf{v}, \Delta\_{b,j-1}\right)}{\chi\_2\left(\mathbf{v}, j-1\right)}, \tag{16}$$

where χ2 v; Δh,j�1 � � ¼ 1 � e �v hx ð Þ ð Þj �h xð Þ j�1 � �=v,

$$g(D\_b g)\_j := \upsilon \frac{g\left(h\left(\mathbf{x}\_j\right)\right) - g\left(h\left(\mathbf{x}\_{j-1}\right)\right)}{1 - e^{-\upsilon \left(h\left(\mathbf{x}\_j\right) - h\left(\mathbf{x}\_{j-1}\right)\right)}}.\tag{17}$$

and h xð Þ is any absolutely continuous complex function on ½ � a; b .

The derivative of a product of functions. The exact finite differences derivative of a product of functions is

$$\begin{split} \left( \left( D\_{f} \text{g} h \right)\_{j} = \frac{\mathbf{g}\_{j+1} h\_{\hat{f}+1} - \mathbf{g}\_{j} h\_{\hat{f}}}{\chi\_{1}(\mathbf{v}, \boldsymbol{j})} = \mathbf{g}\_{j+1} \frac{h\_{\hat{f}+1} - h\_{\hat{f}}}{\chi\_{1}(\mathbf{v}, \boldsymbol{j})} + \frac{\mathbf{g}\_{j+1} - \mathbf{g}\_{j}}{\chi\_{1}(\mathbf{v}, \boldsymbol{j})} h\_{\hat{f}} \\ = \mathbf{g}\_{j+1} \left( D\_{\hat{f}} h \right)\_{j} + h\_{\hat{f}} \left( D\_{\hat{f}} \mathbf{g} \right)\_{j} = e^{-v \cdot \Delta\_{\hat{f}}} g\_{j+1} (D\_{b} h)\_{j+1} + h\_{\hat{f}} \left( D\_{\hat{f}} \mathbf{g} \right)\_{j'} \end{split} \tag{18}$$

where 1 ≤ j < N. Also, for the backwards derivative, we have

$$(D\_b \text{gh})\_j = \mathcal{g}\_j (D\_b \text{h})\_j + h\_{j-1} (D\_b \text{g})\_j = \mathcal{g}\_j (D\_b \text{h})\_j + e^{\nu\_j \Delta\_{j-1}} h\_{j-1} \left( D\_f \text{g} \right)\_{j-1'} \tag{19}$$

where 1 < j ≤ N.

The derivative of the ratio of two functions. For the finite differences, backward derivative of the ratio of two functions we have

$$\begin{split} \left( D\_{b} \mathfrak{z}\_{\overline{h}}^{\mathbb{C}} \right)\_{j} = \frac{1}{\chi\_{2} \left( \upsilon, j-1 \right)} \left( \frac{\mathfrak{z}\_{j}}{h\_{\overline{j}}} - \frac{\mathfrak{z}\_{j-1}}{h\_{\overline{j}-1}} \right) = \frac{1}{\chi\_{2} \left( \upsilon, j-1 \right)} \left( -\frac{\mathfrak{z}\_{j} \left( h\_{\overline{j}} - h\_{\overline{j-1}} \right)}{h\_{\overline{j}} h\_{\overline{j-1}}} + \frac{\left( \mathfrak{z}\_{\overline{j}} - \mathfrak{z}\_{j-1} \right) h\_{\overline{j}}}{h\_{\overline{j}} h\_{\overline{j-1}}} \right) \\ = \frac{(D\_{b} \mathfrak{g})\_{j}}{h\_{\overline{j-1}}} - \mathfrak{g}\_{j} \frac{(D\_{b} h)\_{\overline{j}}}{h\_{\overline{j}} h\_{\overline{j-1}}}, \end{split} \tag{20}$$

$$\left(D\_b \frac{\mathcal{g}}{h}\right)\_j = \frac{(D\_b \mathcal{g})\_j}{h\_j} - \mathcal{g}\_{j-1} \frac{(D\_b h)\_j}{h\_j h\_{j-1}}.\tag{21}$$

summation by parts results can be used in the finding of an appropriate momentum operator for discrete quantum systems. The summation by parts relates two operators between them-

It is advantageous to use a matrix to represent the finite differences derivative on the whole interval so that we can consider the whole set of derivatives on the partition at once. Let us consider the backward and forward exact finite differences derivative matrices Db,f given by

We have used the definition for the backward derivative ð Þ Dbg j for all the rows of the backward derivative matrix Db but not for the first line in which we have instead used the forward

The matrix formulation of the derivative operators allows the derivation of some useful results

Many properties can be obtained with the help of the derivative matrices Db,f . Expressions for the exact second finite differences derivative associated to the exponential function are

obtained through the square of the derivative matrices Db,f . These expressions are

T

1. A similar thing was done for the forward derivative matrix Df . These

∈ CN.

ð28Þ

169

Exact Finite Differences for Quantum Mechanics http://dx.doi.org/10.5772/intechopen.71956

ð29Þ

4. The matrix associated to the exact finite differences derivative

selves and with boundary conditions on the functions.

and

derivative Df g

for the derivative itself.

4.1. Higher order derivatives

matrices act on bounded vectors g ¼ g1; g2; ⋯; gN

$$\left(D\_{\hat{f}}\frac{\mathcal{g}}{h}\right)\_{\hat{f}} = \frac{\left(D\_{\hat{f}}\mathcal{g}\right)\_{\hat{f}}}{h\_{\hat{f}+1}} - \mathbf{g}\_{\hat{f}}\frac{\left(D\_{\hat{f}}h\right)\_{\hat{f}}}{h\_{\hat{f}}h\_{\hat{f}+1}},\tag{22}$$

$$\left(D\_f \frac{\mathcal{S}}{h}\right)\_j = \frac{\left(D\_f g\right)\_j}{h\_j} - g\_{j+1} \frac{\left(D\_f h\right)\_j}{h\_j h\_{j+1}}.\tag{23}$$

Additional properties. A couple of equalities that will be needed below are

$$\frac{1}{\chi\_2(v,\vec{j})} - \frac{1}{\chi\_1(v,\vec{j})} = v,\text{ and }\frac{\chi\_1(v,\vec{j})}{\chi\_2(v,\vec{j})} = e^{v\cdot\Delta\_{\vec{j}}}.\tag{24}$$

For instance, these equalities imply that

$$(D\_{\!\!\!/} \mathbf{g})\_{\!\!\!/} = e^{-v \cdot \Delta\_{\!\!/}} (D\_{\!\!\!/} \mathbf{g})\_{\!\!\!/ + 1}. \tag{25}$$

Summation by parts. An important result is the summation by parts. The sum of equalities (18) and (19) combined with equalities (10) and (11) provide the exact finite differences summation by parts results,

$$\sum\_{j=n}^{m} \chi\_2(\mathbf{v}, \mathbf{j}) \mathbf{g}\_{j+1}(D\_b h)\_{j+1} + \sum\_{j=n}^{m} \chi\_1(\mathbf{v}, \mathbf{j}) h\_j(D\_f \mathbf{g})\_j = \mathbf{g}\_{m+1} h\_{m+1} - \mathbf{g}\_n h\_n \tag{26}$$

where 1 ≤ j < N, and

$$\sum\_{j=n}^{m} \chi\_2(\mathbf{v}, j-1) \mathbf{g}\_j(D\_b h)\_j + \sum\_{j=n}^{m} \chi\_1(\mathbf{v}, j-1) h\_{j-1} \{D\_f \mathbf{g}\}\_{j-1} = \mathbf{g}\_m h\_m - \mathbf{g}\_{n-1} h\_{n-1} \tag{27}$$

where 1 < j ≤ N.

The integration by parts theorem of continuous functions is the basis that allows to define adjoint, symmetric and self-adjoint operations for continuous variables [8, 9]. Therefore, the summation by parts results can be used in the finding of an appropriate momentum operator for discrete quantum systems. The summation by parts relates two operators between themselves and with boundary conditions on the functions.


*2.4. Discussion*

Related works aim to make association, detection, classification, and prediction about mental health. From the analysis of these studies, we identified some open issues about applying passive detection of social situations to support mental health professionals. First, it is necessary to develop solutions able to identify sociability patterns that represent information about social routine of individuals. Second, analyses of such solutions should consider contextual information to identify social situations. Finally, there is a need for solutions that can detect social behavior changes to allow specialized professionals to investigate whether there is a relationship between the identified change and the patient's mental state.

Although the related works aim to identify social behaviors to support mental health monitoring, these studies differ from our solution, so it is a challenge to use objective comparison metrics. For example, some studies develop machine learning models to classify and predict mental states, while our solution aims to extract sociability patterns and detect behavior changes. The works [22–24] also propose solutions capable of detecting sociability patterns, but they differ from our solution. These works design sociability patterns to quantify the duration and frequency of social interactions, while our solution recognizes periods of the day representing individuals' social routine. Besides, these works do not recognize sociability patterns based on contextual data (e.g., weekends, holidays, and rainy days) and do not perform incremental learning, then requiring to execute batch processing.

In comparison to related works, our research has the following contributions. First, this study does not focus on the diagnosis of a specific state or mental disorder but works on identifying situations of interest (i.e., the sociability routine) for mental health professionals. Second, the proposed solution recognizes context-sensitive sociability patterns, so enabling to distinguish normal behavioral variation from behaviors that are considered anomalies. Third, besides identifying the periods of the day when the individual generally socializes, the proposed solution can recognize unusual social routines and significant changes in the patient's social habit. Finally, the proposed solution uses a data stream mining approach to learn from social observations continuously.


**3. Hypopharyngeal cancers with cartilage destruction (T4a N0-N2 disease)**

Patients with cartilage destruction are not ideal candidates for larynx preservation approaches and surgery followed by RT remains the standard of care in this

subgroup. There is less likelihood of complete response with RT or CCRT and reduced success rates due to increased complications of salvage laryngectomy in these patients.