Luciano Da
email: luciano@ifsc.usp.br

F Costa

The Exponential Function: A Mathemagical Hub

The exponential function, including its real-and complex-valued forms, constitutes one of the most important concepts in mathematics. In this work, the exponential function is approached in a progressive manner. After some brief historical remarks, we start our presentation of the exponential function with its conceptual understanding and definition, following with a succession of several related topics including its properties, related functions, statistical aspects, the complex exponential, geometrical structures, the Fourier series and transform, the concept of instantaneous frequency, the generalization of the Fourier transform into the Laplace transform, the matrix exponential, the logarithm of a matrix, linear dynamic systems, discrete-time linear dynamic systems, and Markov chains. A graph obtained by representing the main relationships addressed in the present work then suggests that the exponential function constitutes its respective hub. "... the exponential function is to the product as the identity function is to the sum.

Introduction

The field of mathematics, one of the earliest along the history of science, is characterized by the objective quantification and representation of diverse concepts, structures, and dynamics in terms mainly of values (numbers), relationships between these values (e.g. order or identity), transformations between these values as typically implemented by functions, as well as the organization of these values into structures such as constants, variables, vectors, matrices, graphs, etc.

Complemented by logical and computational resources, the above concepts and constructs have provided most of the basis for scientific modeling, which is primarily aimed at better understanding and predicting the dynamics of diverse real-world world phenomena. In a sense, mathematics can be understood as providing a framework that is employed to represent and simulate entities and dynamics from the real world, as well as a wide range of interesting theoretical abstractions.

As could be expected, the several mathematical con-cepts that have been developed along centuries can often be interrelated. For instance, the integer product can be understood in terms of repeated scalar products and sums, a polynomial on a variable x is composed of terms involving powers of x, and the sine and cosine can be derived from the complex exponential. The existence of these multiple relationships is related to possible ontological frameworks underlying mathematics. For instance, starting with some basic concepts and hypothesis, other mathematical concepts can then be successively derived through respective relationships, such as in the three above examples.

One particular interesting aspect motivated by the development of possible mathematical ontologies concerns the possible identification of more central, or fundamental core concepts from which several other concepts can be related and derived. Interestingly, the possible identification of these core concepts can not only help our understanding of mathematics (and therefore science), but also provide subsidies for respective effective implementations as well as for unifying seemingly diverse mathematical entities.

Needless to say, the identification of possible mathematical ontologies has constituted a substantial challenge when taken in a systematic manner, aimed at incorporating most mathematical results. One example is Hilbert's program, which was aimed at developing a fully compre-hensive framework accounting for the whole of mathematics. Though interesting in itself, this type of approach has been constrained by the identification of Gödel's incompleteness theorems, which showed that there are limits to the self-consistency and formal verification of formal mathematical systems.

The present work develops along the above presented motivation of interrelating mathematical concepts, especially the exponential function in its real and complex forms. However, the approach henceforth described in this work is not aimed to be completely formal or totally comprehensive. Instead, it has a simpler and more pragmatic objective that consists in relating the exponential function to several other mathematical concepts and results. Though we shall present several such interrelationships, it is by no means implied that the exponential function accounts for most of mathematics.

At the same time, as already observed above, the identification of these interrelationships can contribute in several manners to better understanding and applying mathematics. These potential benefits include a more comprehensive appreciation of mathematical concepts, the development of systems (including hardware) founded on effective implementation and calculation of the exponential function, as well as possibly contributing as a didactic resource in which important mathematical results can be more effectively assimilated through the identified interrelationships.

As presented along the present work, the exponential function bears an almost unending number of relationships with other key mathematical concepts and structures. These interrelationships will be here presented in a progressive, successive manner, starting with the description of the exponential function in its real and complex form, followed by the presentation of its relationship with a number of other mathematical concepts including several basic concepts, series, other functions (including powers, logarithms, sine and cosine), difference and differential equations, analytical geometry, number theory, statistics and distributions, dynamic systems, Fourier series and transforms, the Laplace transform, as well as Markov chains.

Some Brief Historic Remarks

The development of the exponential and logarithm concepts extends along a considerable period of time and involved a substantial number of contributors (e.g. [START_REF] Cajori | History of the exponential and logarithmic concepts[END_REF][START_REF] Cajori | A history of mathematics[END_REF][START_REF] Boyer | A history of mathematics[END_REF]).

Here we can only provide an extremely brief summary of some of the respective developments that are somehow more directly related to the subjects to be described in the present work.

Possibly, the first approaches to the subjects of logarithms, also including trigonometric functions, was described by the Scottish mathematician John Napier (1550-1617) in his 1614 work Mirifici logarithmorum canonis descriptio. However, the value e, now known as the Euler constant, was a later contribution by Jacob Bernoulli (1655-1705). In a short period of time, these contributions started being widely adopted as means to facilitate numerical calculations, especially of products, with the help of logarithmic tables. Interestingly, the mechanical device developed by Napier, and known as Napier's bones constitutes a resource for calculation of products and quotients between values that is not based on the concept of logarithms.

After preliminary related developments by the English mathematician Roger Cotes (1682-1716), the important result now amply known as the Euler's formula was described by Leonard Euler (1707-1783) in 1748 in his twovolume work Introduction in anaysin infinitorum. The concepts of logarithm and exponential functions, in particular, contributed substantially for establishing relationships with the concept and calculation of the powers and roots, including concerning complex values, especially thanks to developments by Augustin-Louis Cauchy (1789-1857), in his Cours d'analyze (1821).

The Fourier series was developed mainly by Jean-Baptiste Joseph Fourier (1768-1830) as a means to solve the heat equation (diffusion) on a metal plate, which he described in his reference work Mémoire sur la propagation de la chaleur dans les corps solides (1807).

The development of matrix algebra was to a great extent pioneered by the British mathematician Arthur Cayley (1821-1895), who also employed matrices as resources for addressing linear systems of equations. Cayley focus on pure mathematics included also important contributions to analytic geometry, group theory, as well as in graph theory.

One of the first systematic approaches to the application of matrices to dynamics and differential equations has been developed in the book Elementary matrices and some applications to dynamics and differential equations [START_REF] Frazer | Elementary matrices and some applications to dynamics and differential equations[END_REF], whose first 155 pages present a treatise on matrices, including infinite series of matrices and differential operators. The remainder of the book described the solution of differential equations by using matrices, as well as applications to dynamics of airplanes.

The Arithmetic and Geometric Progressions

A real-valued progression is a sequence of numbers x k indexed by a non-negative value k = 0, 1, 2, . . .. There are two special types of progressions: arithmetic and geometric.

An arithmetic progression is such that:

x k = k∆x + a = x 0 , x 1 , x 2 , . . . (1)
where ∆x, a ∈ R. Observe that x 0 = a.

In other words, each successive element in the sequence can be obtained from the previous by adding a constant amount ∆x.

For instance, in case ∆x = 2 and a = 3, we have:

x k = 2k + 3 = 3, 5, 7, 9, . . . (2)
Given N elements of an arithmetic progression, the respective arithmetic series can be defined as:

N k=0 x k = x 0 + x 1 + x 2 + . . . + x N (3)
The sum of the (N + 1) first elements of an arithmetic progression can be immediately calculated as:

S N = (N + 1)(x N + x 0 ) 2 (4)
So, in the case of the above example, for N = 3, it follows that:

S 3 = 3 + 5 + 7 + 9 = 4(9 + 3) 2 = 24 (5)
A geometric progression can be similarly defined respectively to the product as:

x k = r k a = x 0 , x 1 , x 2 , . . . ( 6
)
where a, r, ∈ R and x 0 = a. Therefore, in a geometric progression each next element is obtained from the previous by multiplying the latter by a fixed ratio r.

As an example, for r = 3 and a = 2 we have:

x k = r k a = 3 k 2 = 2, 6, 18, . . . (7)
As with arithmetic progressions, geometric progressions can be associated respectively to geometric series. The sum of the N + 1 first elements of a geometric progression can be calculated as:

S N = a 1 -r N +1 1 -r = a r N +1 -1 r -1 (8)
for r = 1. In case N → ∞, convergence requires that |r| < 1.

In the case of the example above, for N = 2:

S 3 = 2 + 6 + 18 = 2 3 3 -1 3 -1 = 26 (9)
4 The Exponential Function

The exponential function of a generic real-valued variable x is typically expressed as:

f (x) = exp(x) = e x > 0, x ∈ R (10)
where e = 2.71828 . . . is the Euler number.

Though it is sometimes conceptualized as a function on itself, as suggested by the form exp(x), the exponential function can also be understood as a particular instance of its generalized form:

g(x) = c x , ∀c, x ∈ R ( 11
)
also with g(x) = e x > 0. This function, which shall be henceforth referred to as the general exponential function, should not be confounded with the power function p(x) = x c . Observe that the only difference between Equations 10 and 11 is the fact that the constant c is forced to be equal to e in the former case. The exponential function should not be confounded with the power function, which is defined as:

h(x) = x b (12)
the main difference being that, in the power function the variable x is taken to the power of b, while in the exponential function the Euler constant is taken to the power of x.

Given a generic function, an immediately related aspect concerns its respective inverse f -1 (x). The inverses of the exponential functions correspond respectively to:

f -1 (x) = ln x (13) g -1 (x) = log c x (14)
which applies for b = 1, b, x ∈ R + , where R + is the set of real values x so that x > 0. The functionln() is called the natural logarithm.

Observe that it is the fact that f (x) = e x > 0 for x ∈ R which restricts the domain of inverse of the exponential function to R + .

One interesting manner to look at the exponential function and its logarithm inverse is as providing a bridge to numeric values that are extremely large or small, constituting a kind of telescope/microscope to these realms. For instance we have that: 10 15 = 1.000.000.000.000.000 log 10 (1.000.000.000.000.000) = 15 10 -21 = 0.000000000000000000001

log 10 (0.000000000000000000001) = -21
Given a generic logarithmic function log c (x), the constant c is said to constitutes its respective basis. Therefore, the Napierian logarithm has basis e, with respective logarithm being expressed as:

ln x = log e x (15)
Recall that logarithms can have their base changed according to the following rule:

log c x = log b x log b c = 1 log b c log b x = (constant) log b x (16)
The above results allow us to relate the exponential function e x to its generic counterpart c x . More specifically, for c ∈ R + :

g(x) = c x > 1, ∀x ∈ R =⇒ =⇒ log c g(x) = x = log e g(x) log e c = 1 ln c ln g(x) =⇒ =⇒ ln g(x) = x ln c =⇒ g(x) = e x ln c
=⇒ which leads to:

c x = e x ln c (17)
This is a particularly important relationship that should be kept in mind. In addition, it can be shown that the exponential function satisfies the following relationship:

(e x ) y = (e y )

x = e x y (18)

where x, y ∈ R. An analogue property holds for the general exponential function, i.e.:

(b x ) y = (b y ) x = b x y (19)
It is important to take into account that, in general:

(b x ) y = b (x y ) (20)
which means that the expression b x y is intrinsically ambiguous unless some preliminary convention is assumed regarding the order of the powers.

The identity in Equation 17 also allows us, assuming x ∈ R + , to relate the exponential and power functions as:

x b = e b ln x (21)
Another particularly relevant property, more formally known as group homomorphism (e.g. [START_REF] Mansfield | Background to Set and Group Theory[END_REF][START_REF] Baumslag | Group Theory[END_REF][START_REF] Rose | A Course on Group Theory[END_REF][START_REF] Da | Group theory: A primer. Researchgate[END_REF]) established between the additive and multiplicative binary operations, which is satisfied by both the exponential functions, is as follows:

f (a + b) = f (a) f (b) ⇐⇒ e a+b = e a e b (22)
g(a + b) = g(a) g(b) ⇐⇒ c a+b = c a c b (23)
where a, b, c ∈ R.

The group property also holds for the respective inverse (logarithmic) functions:

ln(a + b) = (ln a)(ln b) (24) log c (a + b) = (log c a)(log c b) (25)
where a, b, c ∈ R + . We can employ the group property, combined with the property in Equation 18, to better understand the operation (e a e b ) c : (e a e b ) c = e a+b c = e c(a+b) = e ca e cb [START_REF] Vetterli | Foundations of signal processing[END_REF] where a, b, c ∈ R. This means that the power by c is distributive respectively to the product of exponentials. Actually, we also have the more general property that:

(a b) c = a c b c (27)
with a, b, c ∈ R. Before the advent of electronic computers, this property that was extensively used for calculating products, which was performed by consulting respective tables of logarithms and calculating the simpler sum instead of the potentially much more demanding original product.

The exponential function also has important features which distinguish it from its more general counterpart. One of them concerns its derivative:

d e x dx = e x (28)
d c x dx = d e x ln c dx = (ln c) c x (29)
for c ∈ R + . Therefore, the exponential function has an intrinsically simpler form than its more general counterpart, the latter requiring a non-unit constant.

The intrinsic properties of the exponential function can be better appreciated by considering graphical constructions such as that shown in Figure 2. Here, we have the exponential function e x in both its continuous form f (x) (blue) as well as being discretized as F k = f (x k ) (red) at equally spaced points x k = x 0 + k∆x, ∀∆x > 0 and k = 0, 1, 2, . . ., along its domain after having started at an arbitrary position x 0 . We have from the property in Equation 22 that:

F k = e x0+k∆x = e x0 e k∆x (30)
which leads to the following interactions:

F 0 = e x0
F k = F k-1 e ∆x = r F k-1 (31)
with k > 0 and r = e ∆x . Equation 31 can be understood as a geometrical progression with ratio r = e ∆x , meaning that each successive element along the discretized exponential can be obtained interactively from the previous instance through a simple real multiplication by r.

The sum of the values of the above progression can be readily calculated by the following expression, which involves the own exponential function:

S = n k=0 F k = F 0 r n+1 -1 r -1 = e x0 e ∆X(n+1) -1 e ∆X -1 (32)
By taking logarithms at both sides of Equation 31, we derive the following interactive equation, which holds irrespectively of the choice of x 0 ∈ R: [START_REF] Nagle | Fundamentals of Differential Equations[END_REF] of which the time-discretized exponential function, by the above developed construction, is intrinsically a solution.

ln (F k ) -ln (F k-1 ) = ∆x
It can be verified that all the above results also hold for ∆x < 0 or r < 0, which can also be understood as mirroring of the exponential function with respect to the y-axis.

We have therefore seen in this section that the exponential function, through its basic property in Equation 22, has the intrinsic characteristic, for equally spaced domain points, of defining a respective geometric progression. This also means that each new value of the discretized exponential can be conveniently obtained through interactive real products with the respective ratio r = e ∆x .

This result means, among other things, that the exponential has a relatively simple construction rule that can be understood in direct analogy with another fundamental mapping, namely the identity function h(x) = x, i.e.:

H k = k∆x + H 0 = H k-1 + ∆x ( 34
)
with H 0 = x 0 and k = 1, 2, . . .. In this sense, it could be informally said that the exponential function is to geometric series as the identity function is to arithmetic series. This would also mean that the exponential function is to the product as the identity function is to the sum.

Inferring the Exponential Function

In the developments described in the previous section, we started with the exponential function being domaindiscretized and then derived the absolute and interactive rules in Equation 22 and 31, respectively. In this section, we will proceed the other way round. More specifically, we will seek for a generic, as yet unknown, function g(x) from which a domain-discrete function G k = g(x k ) can be obtained, with

x k = k∆x + x 0 , x 0 ∈ R, k = 0, 1, . . . ( 35
)
so that G k satisfies the following interactive rule for any x 0 and ∆x:

G k = r G k-1 , ∀x ∈ R, k = 1, 2, . . . ( 36
)
for ∀r ∈ R + and G 0 = g(x 0 ).

G k = r G k-1 , ∀x ∈ R, k = 1, 2, . . ., with G 0 = g(x 0 ).
It follows immediately from Equation 36that:

G k = rG k-1 = r (rG k-2 ) = . . . = r k G 0 (37)
which holds for ∀x ∈ R, k = 1, 2, . . .. We have from Equation 18that:

G k = r k G 0 = e k ln r G 0 (38)
From Equation 35, it follows that:

k = x k -x 0 ∆x ( 39
)
By substituting this identity into Equation 38, we have:

G k = g(x k ) = e ( x k -x 0 ∆x ) ln r G 0 (40)
which constitutes the solution, valid for x k = k∆x+x 0 , of the interactive Equation 36 specifically to x 0 and ∆x. Now, in order the above equation does not depend on ∆x, and considering that we are free to chose any value for r ∈ R+, without loss of generality we can make: r = e ∆x =⇒ ln r = ∆x [START_REF] Da | What is a complex network?[END_REF] which leads to the following possible solution of the original problem:

G k = g(x k ) = e ( x k -x 0 ∆x )∆x G 0 = e x k -x0 G 0 =⇒ =⇒ G k = g(x k ) = e x k G 0 e x0 (42)
with k = 1, 2, . . .. Though we know from the start that g(x 0 ) = G 0 , we still do not have an explicit form for g(x 0 ) because the above equation does not cover the case k > 0. As we are looking for a generic solution g(x) that does not depend on the choice of x 0 , we can take the ansatz G 0 = e x0 , allowing Equation 42 to be rewritten as:

G k = g(x k ) = e x k (43)
As this result does not depend on x 0 or ∆, we obtain the following generic solution to the initially sought function g(x):

g(x) = e x ( 44
)
from which any respectively obtained discrete function G k will satisfy both Equations 36 and 37 which, as developed in Section 4, turns indeed to be the case.

Though in this section we developed the concept of the exponential function while starting from the geometric progression, there are other pathways leading to that same function. For instance, we could start with the question of which is the function that leads to itself under the important operations of differentiation and integration. The answer would be the exponential function. In addition, since:

de x dx = e x (45)
it also follows that the exponential function is infinitely differentiable, i.e.:

d n e x dx n = e x (46)
with n = 1, 2, . . ..

Some Basic Properties

Given a complex (therefore, including real) function f (x) that has infinite derivatives at a point x 0 , its respective Taylor series expansion around x 0 is as follows:

f (x) = f (x 0 ) + ḟ (x 0 ) 1! (x -x 0 )+ + f (x 0 ) 2! (x -x 0 ) + ... f (x 0 ) 3! (x -x 0 ) + . . . ( 47
)
where ḟ is the first derivative, and so on. When x 0 = 0, the Taylor series is often called McLaurin series.

The exponential function f (x) = e x can be expressed in terms of its McLaurin series:

f (x) = exp(x) = e x = = 1 + x + x 2 2 + x 3 6 + x 4 24 + . . . = ∞ k=0 x k k! (48)
with infinite radius of convergence, i.e. ∀x ∈ R, meaning that the exponential function is entire (i.e. holomorphic in the complex plane).

The series above provides a means to numerically calculate the exponential function in terms of the simpler sum and multiplication operations.

The power series for the logarithm function g(x) ln(1 + x) is as follows:

g(x) = ln(x) = = (x -1) - (x -1) 2 2 + (x -1) 3 3 - (x -1) 4 4 + . . . = = ∞ k=1 (-1) k-1 (x -1) k k (49)
which holds for x ∈ (0, 2), therefore not being entire as the exponential function is.

The exponential function has several useful properties. Some of them are presented as follows including, for reference purposes, some already seen in this work¿ e 0 = 1 (50) Now, let us consider the following generic power function of x, also known as a power law :

e x > 0, ∀x (51)
p(x) = a x b (63)
In case x ∈ R + (i.e. a positive real value), the previous equation can be rewritten as:

p(x) = a e ln(x)b (64)
Now, if we take the logarithm at both sides:

ln(p(x)) = ln(a) + ln(x)b = c + b ln(x) (65)
That is why the power law in Equation 63 yields a straight line when represented as a log-log plog.

Some Related Functions

In addition to its several roles in the form g(x) = be ax , the exponential function also integrates several other important functions including, but by no means limited to: sigmoid, gaussian, and gamma. Some of these functions are briefly presented in this section. Given that the gamma function involves the complex form of the exponential function, it will be presented in Section 9.

We start with the following function:

f (x) = 1 -e -ax , x ≥ 0 0, x < 0 (66)
Given that the exponential function g(x) = e ax takes values in the interval (0, 1] for x ≥ 0, the function f (x) above, henceforth called complemented exponential, will also take values in the same interval (0, 1]. Figure 4 Another particularly important function that involves the exponential is the gaussian, which can be written as:

g σ (x) = a e -1 2 ( x-µ σ ) 2 (67)
The parameters µ and σ controls the position of the function along the x-axis and its width, respectively.

Observe that the gaussian can be understood as an exponential whose argument has been taken to the second power, therefore involving the composition of these two functions. Figure 5 depicts a gaussian function being obtained as the composition of a negative parabola into an exponential function. Yet another important function obtained from the exponential function is one of the types of the the sigmoid function, which can be expressed as:

s(x) = 1 1 + e -ax (68)
where the parameter a controls how abrupt the transition from 0 to 1 underwent by this function is.

The Exponential Function in Statistics

Jointly with random variables, the concept of distribution functions plays a key role in probability and statistics (e.g. [START_REF] Degroot | Probability and Statistics[END_REF][START_REF] Kreyszig | Advanced Engineering Mathematics[END_REF][START_REF] Da | Statistical modeling[END_REF][START_REF] Da | Multivariate statistical modeling[END_REF]). More specifically, a function p(x) that satisfies the following requisites can be called a distribution function, or distribution for short:

p(x) ≥ 0, ∀x ∈ R (69) ˆ∞ -∞ p(x)dx = 1 ( 70
)
where X is the name of the associated random variable, whose values are represented as x.

It follows that any function g(x) > 0 with finite area can be, in principle, normalized as a respective distribution by making:

p(x) = g(x) ´∞ -∞ g(x)dx (71)
Given a distribution p(x), its expectancy can be calculated as:

E [X] = ˆ∞ -∞ x p(x) dx ≈ µ ( 72
)
where µ is the average of x as estimated from respective sample. For simplicity's sake, we shall henceforth assume that the average equals the expectancy, i.e. µ = E [X].

The second central moment of p(x), also called variance, can be written as:

Var {x} = E (X -µ) 2 = E X 2 -E [X] 2 ≈ σ 2 (73)
For simplicity's sake, it is henceforth understood that the estimated variance σ 2 is equal to the second central moment, i.e. σ 2 = Var {x}.

The quantity σ = + √ σ 2 corresponds to the standard deviation of X.

As could be expected, the exponential function restricted to x ∈ R + can be found in several important statistical distributions, some of which are briefly described as follows.

We start by the exponential distribution, which can be defined as: One of the most important statistical distributions, if not the most important, constitutes the normal distribution, which can be expressed as:

p α (x) = α e -αx , x ≥ 0 0, x < 0 (74) µ = 1 α ( 75
)
σ 2 = 1 α 2 (76)
g σ (x) = 1 σ √ 2π e -1 2 ( x-µ σ ) 2 (77) (78)
The parameters µ and σ are related to the mean and standard deviation of the respectively associated random variable X. Interestingly, the Dirac delta can be expressed as:

δ(x) = lim σ→0 n σ (x) (79)
this illustrates why the Dirac delta can be understood as having unit area, as this property is preserved as σ → 0.

The Complex Exponential

The incorporation of complex numbers (e.g. [START_REF] Fisher | Complex Variables[END_REF][START_REF] Brown | Complex Variables and Applications[END_REF][START_REF] Kreyszig | Advanced Engineering Mathematics[END_REF][START_REF] Da | Complex numbers: Real applications of an imaginary concept[END_REF]) into mathematics constitutes one of the most important related developments to this area.

It can be shown that the complex exponential adheres to the following power series, which can be understood as its definition:

g(z) = g(x, y) = exp(z) = e z = = 1 + z + z 2 2 + z 3 6 + z 4 24 + . . . = ∞ k=0 z k k! ( 80
)
where and i = √ -1 and x and y are called the real and imaginary parts of the complex number z.

As with the real-valued exponential function, the group property also holds:

g(z) = exp(z) = e z = e x+iy = e x e iy (81)
The complex power function is defined in terms of the complex exponential, respectively to a complex value z and b ∈ R, as:

z b = exp(b ln(z)), for z = 0 ( 82
)
It should be borne in mind that both the complex logarithm and complex power functions are multiple valued (e.g. [START_REF] Fisher | Complex Variables[END_REF][START_REF] Brown | Complex Variables and Applications[END_REF]).

The Euler's formula establishes an important relationship between the complex exponential function and the real-valued sine and cosine functions:

e iθ = cos(θ) + i sin(θ) ( 83
)
This relation can be readily used to express a generic complex number z = x + iy, in terms of its polar representation:

z = x + iy = ρe iθ = ρ [cos(θ) + i sin(θ)] ( 84
)
with ρ ∈ R + and θ ∈ R corresponding to:

ρ = x 2 + y 2 (85) θ = arctan y x ( 86
)
where θ is sometimes understood as the phase of z. In case we have:

u = ρe i(θ+φ) ( 87
)
the term φ can be understood as a phase lag, phase delay or relative phase (e.g. [START_REF] Da | cosine, periodicity, phase, sine[END_REF]).

It is interesting to observe that the quantity ρ = x 2 + y 2 is intrinsically related to the concept of Euclidian distance, which plays a special role in geometric spaces.

The sine and cosine functions can themselves be derived from the complex exponential function as follows:

cos(θ) = e iθ + e -iθ 2 (88) sin(θ) = e iθ -e -iθ 2i (89)
It is also possible to consider the complex exponential defined respectively to a parametrized complex argument z(t) = x(t) + iy(t), with t = [t min , t max ] corresponding to the respective parameter :

exp(z(t)) = e x(t) e iy(t) = e x(t) [cos(y(t)) + i sin(y(t))] (90)
In particular, we can start with:

exp(z(t)) = ρ(t) e iω(t) (91)
and make ρ(t) = 1 and ω(t) = 2πf o t, which leads to the following complex parametric function:

h(t) = e iω(t) (92)
Interestingly, the Euler relationships indicates that this parametric function w(t) corresponds to an helix with amplitude 1 and frequency f 0 along the t-axis. By projecting this complex helix onto the Argand (or complex) plane -with the x-and y-axes corresponding to the real and imaginary portions of the complex values -we get the complex circle with radius 1, which is illustrated in Figure 15.

Figure 10: The complex circle can be understood as the projection onto the Argand plane of the complex parametric function h(t) = e iω(t) , which is periodic with period T 0 = 1/f 0 .

It is interesting to consider the group property of the complex exponential, which accounts for much of its especially important characteristics. Let us start with the following complex values: u = e σu+iθu = e σu e iθu = x u + iy u v = e σv+iθv = e σv e iθv = x v + iy v with σ u , θ u , σ v , θ v ∈ R. It follows that:

u v = e σu+iθu+σv+iθv = e (σu+σv)+i(θu+θv)
The resulting complex value is also a complex exponential whose exponent argument has real part corresponding to the sum of the original correlates of u and v, and imaginary part given as the sum of the respective imaginary correlates.

Let us know separate the complex value v into two respective magnitude and phase parts, i.e.:

v M = e σv = ρ v ∈ R v P = e iθv ∈ C
with σ v , θ v ∈ R. Now we have from the group property that:

u v M = e (σu+σv)+iθu = e (σu+σv) e iθu (93)
Thus, the product of a complex exponential u by a magnitude component v M has as effect the modification only of the real part of the the former, in a manner that is directly analogous to the real exponential. More specifically, this product will change only the magnitude of u.

Similarly, the product of u by the phase component v P yields: u v P = e σu+i(θu+θv) = e σu e i(θu+θv) (94)

implying that only the imaginary part of u is changed, without any alteration of its magnitude ρ u = e σu . More specifically, the phase of u corresponds to the sum of the original phases.

We can therefore conclude that the group property respective to the complex exponential can be understood as two complementary effects corresponding to two complementary effects regarding change of magnitude (analogous to the real exponential) and phase (exclusive to the complex exponential). Now, let us consider the solution of the following equation, implying the complex circle to cross the x-axis at value 1:

e iω(t) = 1 ⇐⇒ ω(t) = 2kπ = 2πf 0 t, k = . . . , -1, 0, 1, . . . ( 95
)
which is satisfied for t = k/f 0 , yielding the period T 0 of the complex parametric function h(t). This equation highlights the intrinsic relationship between the complex exponential and the fundamental constant π.

The gamma function constitutes another example of an important complex function that involves the complex exponential function, through its related complex power form u z-1 . More specifically, we have:

Γ(z) = ˆ∞ 0 u z-1 e -u du ( 96
)
where z is a complex number so that Re(z) > 0, and u is a real-valued integration variable.

In particular, for a non-negative integer value n ∈ R + , we have the following important relationship being established between the gamma function and the factorial function:

Γ(n) = (n -1)! ( 97
)
This relationship illustrates the fact that the gamma function can be understood as a generalization of the factorial function to complex values z|Re(z) > 0.

Figure 11 presents the factorial of non-negative integer values shown as points overlain onto the continuous gamma function Γ(x) restricted to real values x ∈ (0, 5.5]. Observe that, though we have by definition that 0! = 1, this particular case is not covered by the gamma extension of the factorial concept.

Together with the complex exponential function, the gamma function allows us to approach an expression for the following infinite complex series, known as the Riemann zeta function:

ζ(s) = ∞ k=1 1 k s = 1 1 s + 1 2 s + 1 3 s + . . . (98)
where s = σ + it. More specifically, in case σ > 1, we have that the Riemann zeta function can be calculated as:

ζ(s) = 1 Γ(s) ˆ∞ 0 u s-1 e u -1 du (99)
As figured out by Leonard Euler, the Riemann zeta function is related to the concept of prime number (e.g. [START_REF] Da | A first glance at prime numbers[END_REF]), which plays a central role in Number Theory (e.g. [START_REF] Apostol | Introduction to Analytic Number Theory[END_REF][START_REF] Vidogradov | Elements of Number Theory[END_REF]), through the following relationship:

ζ(s) = p is prime 1 1 -p s (100)
Therefore, we have that the complex exponential function is also closely related to fundamental mathematical concepts such as those studied in Number Theory.

Exponential Geometry

The exponential function in its complex form can be associated to parametric curves (e.g. [START_REF] Carmo | Differential geometry of curves and surfaces[END_REF][START_REF] Da | What can curvature tell us about shape? Researchgate[END_REF]) in respective geometrical spaces.

A parametric curve γ(t) can be understood as the mapping of a continuous parameter t ∈ [a, b] into a continuous succession of points (x, y) in the R 2 space. More specifically, we can write:

γ(t) : t ∈ [a, b] -→ γ(t) = (x(t), y(t)) (101)
Given a complex function g(z(t)) = x(t) + iy(t), we can always make the following bijective association: g(z(t)) ←→ γ(t)

x(t) + iy(t) ←→ (x(t), y(t))

(102)

For instance, in the case of the helix h(z(t)) = h(t) = e i2πf0t , we would have:

h(z(t)) ←→ γ(t) cos(2πf 0 t) + i sin(2πf 0 t) ←→ (cos(2πf 0 t), sin(2πf 0 t)) (103)
Figure 12 illustrates this function for t ∈ [0, 4] and f 0 = 2.

Figure 12: The complex helix h(t) = e i2πf 0 t represented as a parametric curve in the R 2 × t space for t ∈ [0, 4] and f 0 = 2. Now, consider the complex parametric function g(z) = ρ(t)e ω(t) , with:

ρ(t) = e -at (104)
ω(t) = ω(t) = 2πf 0 t (105)
This yields a parametric curve in R 2 × t which is modulated by the varying amplitude corresponding to a decaying exponential ρ(t) = e -1t , as shown in Figure 13 respectively to a = 1 and f 0 = 2.

Figure 14 depicts the same function as in the previous example, but now projected into the R 2 plane.

Given a parametric curve, it is possible to define its respective derivatives for k = 1, 2, . . . as:

d k γ(t) dt k = d k x(t) dt k , d k y(t) dt k (106)
As an example, in the case of the helix in Equation 103, we would have:

d γ(t) dt = 2πf 0 (-sin(2πf 0 t), cos(2πf 0 t)) (107)

The Fourier Series and Transform

Let g(t) be a real (or complex) function with period T . Then, in case its Fourier Series (e.g. [START_REF] Brigham | Fast Fourier Transform and its Applications[END_REF][START_REF] Phillips | Signals, Systems and Transforms[END_REF][START_REF] Da | Shape Classification and Analysis: Theory and Practice[END_REF]) exist, it can be expressed as:

c k = 1 T ˆT g(t) exp (-i2πkt) dt (108) g(t) ≈ N k=-N c k exp i 2πkt T ( 109
)
where c k are often called the Fourier coefficients respectively to the represented function g(t). The complex Fourier series in Expression 109 provides a means to approximate (and sometimes fully recover) the original function g(t) in terms of a linear combination of its respective coefficients.

The Fourier transform (e.g. [START_REF] Brigham | Fast Fourier Transform and its Applications[END_REF][START_REF] Da | Shape Classification and Analysis: Theory and Practice[END_REF]) of a nonnecessarily periodic function g(t), as well as its inverse, in case they exist, can be calculated as:

G(f ) = F {g(t)} = ˆ∞ -∞ g(t) exp(-i2πf t)dt (110) g(t) = F -1 {G(f )} = ˆ∞ ∞ G(f ) exp(i2πf t)df (111)
Given two complex functions g(t) and h(t), their Hermitian product can be expressed as:

g(t), h(t) = ˆ∞ -∞ g(t) h * (t)dt (112)
It is interesting to observe an intrinsic analogy between the coefficients c k in the Fourier series and the Fourier transform G(f ). In fact, both these expressions can be rewritten in terms of the Hermitian inner product as:

c k = 1 T g(t), exp (i2πkt) (113) G(f ) = g(t), exp(i2πf t) (114)
As the real-valued inner product, its Hermitian counterpart can be understood as a measurement of the similarity between its two arguments, however influenced by the magnitudes of the respectively compared functions (e.g. [START_REF] Da | Convolution! Researchgate[END_REF]). In this sense, the Fourier series and transform can be understood as approximating the original function by linearly combining complex exponentials with diverse frequencies with weights corresponding to the similarity between each of those components and the original function g(t).

Of particular interest is the fact that the basic components used for the linear combinations involved in both the Fourier series and transform correspond to complex exponential. The effectiveness of this basis for representation of functions stems from the fact that these complex exponential components are orthogonal one another.

As a Fourier transform calculation example, given the truncated (or causal) exponential function:

g(t) = e -σt , t ≥ 0 0, t < 0 ( 115
)
with σ ≥ 0: Let us calculate its respective Fourier transform:

G(f ) = ˆ∞ 0 e -σt e -i2πf t dt = = ˆ∞ 0 e -(σ+i2πf )t dt = = -1 σ + i2πf e -(σ+i2πf )t ∞ 0 = 1 σ + i2πf = 1 s (116) with s = σ + i2πf = σ + iω.
By separating the real and imaginary part, we obtain:

1 σ + iω = σ σ 2 + ω 2 -i ω σ 2 + ω 2 (117)
When the direct and inverse Fourier transforms of a function g(t) exist, we can write the respectively defined Fourier transform pair as:

g(t) ←→ G(g) (118)
Some examples of these pairs include:

e i2πf0t ←→ δ(f 0 ) (119) cos (2πf 0 t) ←→ 0.5 [δ(-f 0 ) + δ(f 0 )] (120) sin (2πf 0 t) ←→ 0.5i [δ(-f 0 ) -δ(f 0 )] ( 121
)
where δ(f 0 ) = δ(f -f 0 ) is the Dirac delta function shifted to the position f 0 .

Of particular interest is the Fourier transform property:

d β g(t) dt β ←→ (i2πf ) β G(f ) (122)
which not only allows the generalization of the derivative to complex orders β ∈ C, but also establishes an intrinsic relation between the complex exponential, through its power form, and the derivative of complex functions.

12 The Discrete Fourier Transform -DFT Let us start with the complex exponential component with period T 0 = 1/f 0 :

h(t) = e -i2πf0t (123)
By quantizing the time parameter t as t = k∆t, k = 0, 1, . . . , N -1, with ∆t = T0 N = 1 f0N :

h 1,k = h(t = k∆t) = e -i 2πk N (124)
Now, we can define additional complex exponential components h j with periods T j = T 0 /j that are integer fractions of T 0 , which leads to f j = jf 0 and f j = j/T 0 . We can now write:

h j,k = e -i2πfj t = e -i2πjf0t = e -i 2πjk N (125) for j = 2, . . . , N -1.
Jointly with the constant component h 0,k = 1, the above components h j,k constitutes the basis of the discrete Fourier transform, which can be combined into an N × N matrix W N .

Figure 15 illustrates the DFT basis element h 1,k for N = 4. The DFT matrix for N = 4 corresponds to:

W 4 =     ← h 0,k → ← h 1,k → ← h 2,k → ← h 3,k →     =     1 1 1 1 1 -i -1 i 1 -1 1 -1 1 i -1 i     (126)
Observe that:

W 4 = W T 4 (127)
which is verified for any N . The DFT G of an N-elements time-discrete function g(t), represented in terms of a respective vector g, as well as its respective inverse (IDFT), can now be respectively expressed in terms of the following matrix equations (e.g. [START_REF] Brigham | Fast Fourier Transform and its Applications[END_REF][START_REF] Phillips | Signals, Systems and Transforms[END_REF][START_REF] Vetterli | Foundations of signal processing[END_REF][START_REF] Da | Signals: From analog to digital, and back[END_REF]):

G = W N g (128) g = W -1 N G (129)
These two expressions, which stem directly from the respective Equations 108 and 110, can again be understood in terms of Hermitian inner products between the components of the Fourier basis (i.e. the complex exponential e i2πf t ) and the time-discrete signal g to be analysed/transformed. More specifically, each element j of G is obtained by multiplying the respective row j of W N (which is the conjugate complex of the Fourier components, as required by the Hermitian inner product) and the discrete-time signal g = [g k ] = g(k∆t).

Given a complex matrix A, it is said to be unitary if and only if:

A -1 = (A T ) * = (A * ) T (130)
In the case of the DFT matrix W 4 , it follows that:

W 4 W T 4 * = W 4 [W 4 ] * = (131) =     1 1 1 1 1 -i -1 i 1 -1 1 -1 1 i -1 i         1 1 1 1 1 i -1 -i 1 -1 1 -1 1 -i -1 -i     = (132) =     4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4     = 4 I (133)
from which it is said that W 4 is a quasi-unitary matrix. It can be verified that this property applies to any nonnegative integer value N , i.e.:

W N W T N * = W N [W * N ] T = W N W * N = N I (134)
The fact of W N being a quasi-unitary and symmetric matrix allows the effective determination of its inverse as:

W N [W N ] * = N I =⇒ (135) =⇒ [W N ] * = N W -1 N I = N W -1 N =⇒ (136) =⇒W -1 N = 1 N [W N ] * (137)
Therefore, we can calculate the DFT and IDFT as:

G = W N g (138) g = 1 N [W N ] * G (139)
13 Instantaneous Frequency

The concept of instantaneous frequency (e.g. [START_REF] Boashash | Estimating and interpreting the instantaneous frequency of a signal: Part 1[END_REF][START_REF]Wikipedia. Instantaneous phase and frequency. Wikipedia, The Free Encyclopaedia[END_REF][START_REF] Da | Instantaneous signal analysis[END_REF]) provides and interesting and useful extension of the more traditional concept of frequency of a signal. The definition of the instantaneous frequency of a signal s(t), a purely real function of an independent variable f having S(f ) as its respective Fourier transform, relies on the respective analytical representation of s(t) as:

s a (t) = F -1 {2 h(f )S(f )} (140)
where h(f ) is the heaviside function:

h(f ) =    0, f < 0 1/2, f = 0 1, f > 0 (141)
The instantaneous amplitude and instantaneous phase of s(t) can now be defined as:

A i (t) = |s a (t)| φ i (t) = arctan Im(sa(t))
Re(sa(t)) (142) from which the instantaneous frequency of s(t) can be expressed as:

f i (t) = 1 2π d dt φ i (s(t)) (143)
As an example of calculation of the instantaneous frequency of a function, consider g(t) = cos(2πf 0 t). First, we obtain the respective analytical representation as:

g a (t) = F -1 {2 h(f )G(f )} = F -1 0.5δ(f 0 ) = 0.5e i2πf0t (144)
from which the respective instantaneous phase can be obtained as:

φ i (t) = arctan Im(s a (t)) Re(s a (t)) = sin(2πf o t) cos(2πf 0 t) = 2πf 0 t (145)
leading to the following instantaneous frequency:

f i (t) = 1 2π d dt φ i (s(t)) = 1 2π d dt 2πf 0 t = f 0 (146)
Therefore, as could be expected, the cosine function g(t) = cos(2πf 0 t) has instantaneous frequency f 0 at any value of t ∈ R. However, the instantaneous frequency of more general signals will often yield varying values (e.g. [START_REF] Da | Instantaneous signal analysis[END_REF]), being closely related to the concept of amplitude modulation -AM, as used in telecommunications.

The Laplace Transform

Given a complex function g(s) of a complex variable s, its respectively Laplace transform (e.g. [START_REF] Da | The Laplace transform in a nutshell[END_REF]) can be defined as follows:

L {f (t)} (s) = F (s) = ˆ∞ 0 g(s) e -st dt (147)
where

s(t) = σ + iω, with σ, ω ∈ R.
Observe that the Laplace transform basis components can be rewritten as:

e -st = e -σt e -iωt (148)
thus indicating that the Laplace transform constitutes a generalization of the Fourier transform incorporating components with exponential decay e -σt , which contributes to the effective representation of a wider range of possible signals.

In case g(s) = g(t) is a function of a a non-negative real argument t so that:

g(t) = g(t), t ≥ 0 0, t < 0 (149)
we can write:

L {g(t)} (s) = ˆ∞ 0 g(t) e -σt e -iωt dt = = ˆ∞ 0 g(t) e -σt e -iωt dt = = ˆ∞ -∞ g(t) e -σt e -iωt dt = = F g(t) e -σt (f ) (150)
As an example, let us calculate the Laplace transform of the truncated exponential function:

g(t) = e -at , t ≥ 0 0, t < 0 (151)
By applying the definition:

L {f (t)} (s) = ˆ∞ 0 e -at e -st dt = = ˆ∞ 0 e -(a+s)t dt = = -1 a + s e -(a+s)t ∞ 0 = 1 s + a (152)
for Re(s) < -a.

Similarly to the Fourier transform, the Laplace transform is also directly related to the operations of differentiation and integration:

ˆt 0 f (t)dt ←→ F (s) s (153) df (t) dt ←→ sF (s) -f (0) (154)
15 Ordinary Differential equations

Differential equations are at the core of physical modeling, as they provide an effective representation of several phenomena in terms of instantaneous properties such as position and velocity (e.g. [START_REF] Da | Modeling: The human approach to science[END_REF]). Among the several types of differential equations, those called ordinary differential equations -ODEs -are frequently adopted in modeling approaches. Basically, this type of differential equations is characterized by the involved unknown elements being a function of a single variable, such as time or position.

In the present work, we shall focus on linear ordinary differential equations, which are characterized by satisfying the superimposition principle (e.g. [START_REF] Nagle | Fundamentals of Differential Equations[END_REF]). In particular, consider the following linear ODE:

ġ(t) = a g(t) (155)
g(t = t 0 ) = g 0 ( 156
)
which means that the tendency (or velocity) of the increase of the variable g(t) is proportional, through t the constant a ∈ R, to its current value. In the case of a > 0, we would have that the quantity g(t) tends to increase at a rate dg(t)/dt proportional to its current value. The initial condition, g(t = t 0 ) = g 0 , informs the value of the variable g(t) at an initial time instant t 0 .

Let us solve this equation as follows: Thus, the solution of the original differential equation can be expressed as: g(t) = e ln(g0)-at0 e at =⇒ =⇒ g(t) = g 0 e -at0 e at =⇒ =⇒ g(t) = g 0 e a(t-t0) (159)

dg(t) dt = a g(t) =⇒ =⇒ 1 g(t) dg = a dt =⇒ =⇒ ˆ1 g(t) dg = ˆa dt =⇒
Therefore, we have that the solution of the possibly most basic linear ordinary differential equation involves the exponential function. In other words, the exponential function is such that its derivative is proportional to itself, therefore constituting a kind of eigenfunction associated to the original differential equation 155.

Interestingly enough, as we will see in Section 19, an analogue solution can be found for a system of linear ordinary differential equations. However, it is necessary to get acquainted with the useful concept of matrix exponential as a preparation for that, which provides the subject for the following section.

Exponential Directional Field

Given an ordinary differential equation on a single variable x, the concept of directional field provides a particularly interesting resource for visualizing, and better un-derstanding, the characteristics of the given differential equation as well as its possible solutions.

Let us consider the following linear ODE:

dx(t) dt = ẋ(t) = x(t) (160)
Its respective directional field can be obtained by plotting a short line segment having inclination corresponding to the derivative ẋ(t, x) at each of a set of respective discrete lattice points (t, x) as determined by the original differential equation, therefore defining a respective tangent field. Figure 16 Figure 16: A directional field, corresponding to the set of green line segments, obtained for Eq. 160. The slope (inclination) of each of these segments corresponds to derivative ẋ(t, x) of x(t) at each of the considered discrete points (t, x). Given an initial condition (starting points at the left-hand side of the plot), the respective solution can be obtained by following the tangent field.

Observe that, in the case of this particular ODE, the derivatives ẋ(t, x) at the points (t, x) do not depend of t, but only on x, in which case the differential equation is said to be autonomous.

Also shown in Figure 16 are several possible solutions of the considered differential equation, obtained respectively to each initial conditions specified by the left-hand points. These solutions are obtained by following the tangent field as one moves along the horizontal axis.

The solution shown in gray, respective to the initial condition (x(t = -1) = 0), corresponds to the only equilibrium solution of the considered ODE. More specifically an equilibrium solution is such that ẋ = 0 for any value of t. In the case of this particular ODE, the only equilibrium solution can be said to be unstable, since any small perturbation along its trajectory will unavoidably cause it to either increase or decrease, which it will do in exponential manner.

As illustrated by the above example, the directional field of a given ODE provides a direct visualization of all possible solutions in the considered window. Additional examples of directional fields, also including systems of two differential equations, can be found in [START_REF] Da | Visualizing the content of differential equations[END_REF].

The Matrix Exponential

In a direct analogy to Equation 48, it is possible to express the exponential of a complex matrix A, with dimension N × N in terms of the following power series:

e A = 1 + x + A 2 2 + A 3 6 + . . . = ∞ k=0 A k k! (161)
Observe that the exponential of the matrix A is itself a matrix with the same dimension as the matrix A from which it derives.

As an example, let:

A =     0 1 0 1 1 0 1 0 2 2 2 -1 0 0 0 3     (162)
From Equation 161, we have that: where ≈ stands for the fact of presenting truncated values in the result, and/or taking a finite number of power series terms.

e A ≈     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1     +     0 1 0 1 1 0 1 0 2 2 2 -1 0 0 0 3     + . . . = =     2
As could be expected, the matrix exponential and the logarithm of a matrix (to be briefly addressed in Section 18) are related as:

b A = e A ln b (163)
The following property is observed by the matrix exponential:

e (A T ) = e A T (164)
In the case of two complex square matrices A and B with the same dimension, we also have that:

e A+B = e A e B (165)
which leads to:

e A e -A = e A-A = I (166) from which we have that the inverse of matrix e A is:

e A -1 = e -A (167)
Interestingly, the matrix corresponding to the matrix exponentiation e A is always invertible.

The Logarithm of a Matrix

Let A be a complex matrix. In case it is invertible, it is possible to obtain its respective logarithm in terms of the following power series:

g(A) = ln(A) = = (A -I) - (A -I) 2 2 + (A -I) 3 3 -. . . = = ∞ k=1 (-1) k-1 (A -I) k k (168)
We have that:

e ln(A) = A (169)
The properties of the logarithm of matrix are relatively more intricate than those of the logarithm function, often involving additional requirements as matrices being positive definite, etc.

Linear Dynamic Systems

Linear dynamic systems are characterized by obeying the superimposition principle. More specifically, if x and y are solutions of a given such type of system, x + y will also be a solution.

Linear dynamic systems are often represented in terms of respective systems of (in our case ordinary) differential equations, such as the following system of homogeneous differential equations:

       ẋ1 = a 1,1 x 1 + a 1,2 x 1 + . . . + a 1,N x N ẋ2 = a 2,1 x 1 + a 2,2 x 1 + . . . + a 2,N x N . . . ẋN = a N,1 x 1 + a N,2 x 1 + . . . + a N,N x N (170)
which can be directly translated to the following matrix equation:

Ẋ(t) = AX(t) (171)
X(t 0 ) = X 0 (172)
with:

A =     a 1,1 a 1,2 . . . a 1,N a 2,1 a 2,2 . . . a 2,N . . . a N,1 a N,2 . . . a N,N     (173)
As with the linear ODE ġ(t) = ag(t) seen in Section 15, the solution of the above system can be obtained as: In case an N × N matrix A is symmetric and has N distinct real-valued eigenvalues λ i associated to respective eigenvectors v i (e.g. [START_REF] Sagan | Boundary and eigenvalue problems in mathematical Physics[END_REF][START_REF] Da | Eigenvalues, Eigenvectors. Researchgate[END_REF]), the solution (e.g. [START_REF] Nagle | Fundamentals of Differential Equations[END_REF]) of the respectively defined homogeneous system of linear differential equations can be written as:

x(t) = e A(t-t0) x 0 (174)
x(t) = = c 1 e λ1(t-t0) v 1 + c 2 e λ2(t-t0) v 2 + . . . + c N e λ N (t-t0) v N (176)
where:

     c 1 c 2 . . . c N      =   ↑ ↑ . . . ↑ v 1 v 2 . . . v N ↓ ↓ . . . ↓   -1      x 0,1 x 0,2 . . . x 0,N      (177)
20 Discrete-Time Linear Dynamic Systems

Given a homogeneous linear dynamic system as in the previous section (Eq. 170), it is possible to write its (possibly approximate) interactive finite difference (order 1) equation as follows:

             x [t+∆t] 1 = a 1,1 x [t] 1 + . . . + a 1,N x [t] N ∆t + x [t] 1 x [t+∆t] 2 = a 2,1 x [t] 1 + . . . + a 2,N x [t] N ∆t + x [t] 2 . . . x [t+∆t] N = a N,1 x [t] 1 + . . . + a N,N x [t] N ∆t + x [t] N
assuming that the time values t are organized as t [k+1] = t [k∆t+t0] . In addition, the initial state is represented as X 0 = X [t=t0] .

The above finite difference approximation can be obtained simply by considering that:

dx(t) dt = lim ∆x→0 x(t + ∆t) -x(t) ∆t ≈ x(t + ∆t) -x(t) ∆t ( 178
)
where tha approximation assumes small values of ∆t.

The system of interactive equations in Equation 178, also called a map, can be compactly expressed in terms of the following matrix equation:

x [t+∆t] = A x [t] ∆t + x [t] = (A∆t + I) x [t]
(179)

x [t0] = x 0 (180)
By making C = (A∆t + I):

x

[t+∆t] = C x [t] (181)
x

[t0] = x 0 (182)
Which has the following solution:

x [t] = e ln C (t-t 0 ) ∆t x 0 = C t-t 0 ∆t

x 0 (183)

Markov Chains

In case the homogeneous system of linear ODEs in the previous section has ∆t = 1 and t 0 is an integer value, we can write:

x [t+1] = A x [t] + x [t] = (A + I) x [t] (184)

x [t0] = x 0 (185)
By making B = A + I, we get the mapping:

x [t+1] = B x [t] (186)
x [t0] = x 0 (187)

In case the square matrix B is a stochastic matrix, having non-negative entries so that the sum of respective columns are all equal to 1, the above mapping can be understood as a Markovian chain (e.g. [START_REF] Norris | Markov Chains[END_REF][START_REF] Kemeny | Finite Markov Chains[END_REF]), with the following solution:

x [t] = e ln B(t-t0) x 0 = B t-t0 x 0 (188)

A stochastic matrix can be shown always to have an eigenvalue equal to 1. A Markov chain associated to an irreducible matrix B will converge to an equilibrium (or stationary) state given by the respective eigenvector associated to the eigenvalue 1. The matrix B will be said to be irreducible when the respectively derived digraph is not strongly connected, i.e. it is possible to reach any of its nodes through paths starting at any other possible node.

In case matrix P can be diagonalized, or eigendecomposed, we can write:

V B = V Λ =⇒ (189) =⇒ B = V ΛV -1 ( 190
)
where Λ is a diagonal matrices containing the eigenvalues λ i of B in descending order and V is the matrix composed having the respective eigenvectors v i as columns.

We have the following interesting property: This yields the following discrete-time solution of the Markov chain associated to B:

B n = V Λ (V -1 V ) Λ (V -1 V ) Λ . . . Λ (V -1 V ) ΛV -1 = = V [Λ] n V -1 (
x [t] = a 1 λ t 1 v 1 + a 2 λ t 2 v 2 + . . . + a N λ t N v N ( 193
)
where the constants a i ∈ R are determined from the initial state x [0] . This interesting result confirms that the equilibrium dynamics is defined by the eigenvector v 1 associated to the eigenvalue with maximum value (λ 1 = 1). It is also of interest to compare the equation above with that in Equation 176.

Concluding Remarks

In addition to its own theoretical importance, mathematics has provided the main foundation for modeling in the physical sciences. To a great extent, this key role has been allowed by the intrinsic potential of mathematical concepts and structures to represent entities and phenomenal from the physical world.

In this work, after developing the concept of the exponential function, we successively presented its direct implications and relationships with several important, or even central, mathematical concepts ranging from fundamental constants to the operations of complex differentiation and integration, while being related even to number theoretical concepts as prime numbers. Figure 18 presents a graph, or network, obtained by representing several of the concepts addressed in the present work as nodes, while many of their interrelationships have been indicated by respective links. Despite its relatively recent history, the exponential and its logarithm inverse are related to an impressive range of key mathematical concepts. For instance, several of the fundamental mathematical functions, shown within the green box in Figure 18, are directly related to the complex exponential function. In addition, several important functions can be derived from the exponential function, as well as the Fourier and Laplace transforms, which allow important connections with the operations of complex differentiation and integration. Thus, in a sense, a wide range of key mathematical concepts can be directly associated to the exponential function, which is reflected in the large number of connections attached to the three respective nodes and, in particular, to the complex exponential function, which can be identified as the main hub (i.e. node with the highest degree, or number of connections, e.g. [START_REF] Newman | Networks: An introduction[END_REF][START_REF] Barabási | Network Sience[END_REF][START_REF] Da | What is a complex network?[END_REF]) of the graph in Figure 18.

Though the present work focused attention on mathematical and statistical concepts related to the exponential function, the latter has also plays a central role in respective applications to a wide range of areas, ranging from electrical engineering to data science, which is planned to be covered in subsequent works.

Figure 1

 1 illustrates several examples of the general exponential functions obtained for equally spaced values of c centered at c = e. The exponential function for c = e is highlighted in red.

Figure 1 :

 1 Figure 1: Several instances of the general exponential function obtained for 7 values of the constant c equally distributed (between -2 and 2) around the value c = e. The exponential function for c = e is highlighted in red.

Figure 2 :

 2 Figure 2: The continuous exponential function f (x) = e x and a possible respective discretization F k (red points) defined by abscissae values x k = x 0 + k∆x, ∆x > 0 and k = 0, 1, 2, . . .

F 1 =F 2 =F 3 =

 123 e x0+∆x = e x0 e ∆x = F 0 e ∆x e x0+2∆x = e x0 e ∆x+∆x = F 1 e ∆x e x0+3∆x = e x0 e 2∆x+∆x = F 2 e ∆x . . .

Figure 3 :

 3 Figure 3: A generic function g(x) (in blue) so that its values G k (in red) follow a geometric progression, i.e. G k = r G k-1 , ∀x ∈

e

  ln x = ln (e x ) = x, ∀x (54) e α(a+b) = e αa e αb , ∀α, a, b ∈ R (55) ln(ab) = (ln a) + (ln b), ∀a, b, ∈ R + (56) e x ln b = b x , ∀x ∈ R, b ∈ R + (57) e ab = e b a = (e a ) b , ∀a, b ∈ R (58) de x dx = e x , ∀x (59) ˆx -∞ e x dx = e x , ∀x (60) An interesting graphical property of the exponential function can is as follows. Let us start with the generic exponential function: g(x) = a e bx (61) with a, b ∈ R. If we take the natural logarithm at both sides, we obtain: ln(g(x)) = h(x) = ln a e bx = ln(a) + bx (62) which corresponds to a linear function h(x) on x having intercept equal to ln(a) and slope b. That is why the semilog plot of the generic exponential function in Equation 61 will necessarily yield a straight line.

  illustrates this function for several values of a.

Figure 4 :

 4 Figure 4: The complemented exponential function for seven values of the parameter a, centered at a = e. The higher the value of a, the most abrupt the transition from 0 to 1.

Figure 5 :

 5 Figure 5: Graphical representation of obtaining a Gaussian function g(x) as the composition of the negative parabola -0.5x 2 into the exponential function f (x) = exp(x).

Figure 6

 6 Figure 6 illustrates the gaussian function for a = 1, µ = 1, and several values of σ. It should be kept in mind that the area of the gaussian function is not necessarily equal to 1, as is the case with the closely related normal distribution to be addressed in Section 8.

Figure 6 :

 6 Figure 6: The gaussian function for a = 1, µ = 1, and several values of σ.

Figure 7 :

 7 Figure 7: One of the types of sigmoid function for several values of its parameter a, which controls how abrupt the transition from 0 to 1 is.

Figure

  Figure reffig:expons illustrates several instances of the exponential distribution respectively to varying values of its parameter α.

Figure 8 :

 8 Figure 8: Examples of several exponential distributions for varying values of its parameter alpha.

Figure 9 :

 9 Figure 9: Examples of several normal distributions assuming µ = 1 and varying the parameter σ. All these distributions have area equal to 1. It is interesting to compare these distributions with the respective unnormalized versions in Fig. 6.

Figure

  Figure reffig:normals shows several instances of the normal distribution respectively to µ = 1 and varying values of its parameter σ.Interestingly, the Dirac delta can be expressed as:

Figure 11 :

 11 Figure 11: The gamma function Γ(x) restricted to real values x along the interval (0, 5.5] and respectively overlain integer-valued factorials (points) of non-negative integers n = 1, 2, . . . considering x ∈ (0, 5.5]. This construction emphasizes the fact that the gamma function can be used to generalized the factorial function to continuous values (and also to complex numbers).

Figure 13 :

 13 Figure 13: The parametric function defined by the complex parametric function g(z) = ρ(t)e ω(t) for a = 1 and f 0 = 2.

Figure 14 :

 14 Figure 14: A spiral is obtained by projecting the parametric curve of Fig. 13 into the Argand plane (associated to the R 2 plane).

Figure 15 :

 15 Figure 15: The basis elements h 1,k , with k = 0, 1, 2, . . . N -1 for N = 4.

  =⇒ ln(g(t)) = a t + c =⇒ =⇒ e ln(g(t)) = e a t+c =⇒ =⇒ g(t) = e c e at (157) Observe that this general solution depends on the constant c ∈ R, which is not yet determined. By taking into account the initial condition, we can write: =⇒ g(t 0 ) = g 0 = e c e at0 =⇒ =⇒ e c = g 0 e -at0 =⇒ =⇒ c = ln g 0 e -at0 =⇒ =⇒ c = ln (g 0 ) -at 0 (158)

  illustrates a directional field (set of green line segments) obtained for the above equation considering t ∈ [-1, 1] and x ∈ [-1, 1].

Figure 17

 17 Figure 17 presents the solution of the system of linear, homogenous differential equations specified by the following matrix and considering x(0) = [-1 1 0 2] T for t ∈ [0, 5]:     -0.2 -0.4 -0.1 0.2 -0.4 -0.6 -0.3 -0.1 -0.1 -0.3 0.4 -0.1 0.2 -0.1 -0.1 -0.6    

Figure 17 :

 17 Figure 17: Solution of the system of linear, homogenous differential equations specified by the matrix in Eq. 175, obtained by using Eq. 174, considering x(0) = [-1 1 0 2] T adn t ∈ [0, 5].

Figure 18 :

 18 Figure 18: A graph illustrated some of the arithmetic, algebraic and functional relationships between the exponential function and several other mathematic concepts. Five fundamental constants, as well as the infinity are shown within the green box The Gaussian function is shown as gs(), the sigmoid function as s(), and the Fourier series (or transform) as F . The zeta function is shown as ζ, the Laplace transform as L, differentiation as ∂ and integration as ´. Complex values and functions are shown as blue nodes, and z and u are complex variables, while b and x are a real variables. The node associated to the complex exponential function e z constitutes the main hub in the presented graph. Arguably, the exponential function directly relates to a considerable breadth of mathematical concepts.

Acknowledgments

Luciano da F. Costa thanks CNPq (grant no. 307085/2018-0) and FAPESP (grant 15/22308-2).

Observations

As all other preprints by the author, the present work contains preliminary work subject to further revision and validation. Respective modification, commercial use, or distribution of any of its parts are not possible. Many of the preprints by the author are also available in Hal and/or arXiv. This work can also be cited by using the DOI number or article identification link. Thanks for reading.


Sagar Verma

Granular Ai

Siddharth Gupta

Kavya Gupta
email: kavya.gupta100@gmail.com

Aligning Geospatial AI for Disaster Relief with The Sphere Handbook

The Sphere handbook and its core premise of right to life with dignity have been broadly adopted, establishing a standard operating procedure for global humanitarian intervention. Plenty of machine learning methods aim to aid in disaster relief. While performing exceptionally on a machine learning task, these methods fail to deliver targeted effort to the victims of natural disasters. We argue that this is due to the misalignment of such methods with real-world relief practices. This paper presents the alignment of the Sphere guidelines with Geospatial AI solutions. We show several limitations in different machine learning methods proposed for disaster relief in recent years. We take the case of WASH requirements during flood disasters, extend these models to align with Sphere guidelines, and build a solution that has a much better potential to serve individuals stuck in disasters.

Application Context

Natural hazards have the potential to cause catastrophic damage and significant socioeconomic loss. The actual damage and loss observed in the recent decades have shown an increasing trend. As a result, disaster managers must take a growing responsibility to protect their communities by developing efficient management strategies. Some set principles or guidelines should govern these strategies. The Sphere Handbook is such a guideline coming out of The Sphere Project1 based on two core beliefs quoted as:

• People affected by disaster or conflict have the right to life with dignity and, therefore, the right to assistance; and

• All possible steps should be taken to alleviate human suffering arising out of disaster or conflict.

In the last few years, we have seen rapid advances in using Geospatial AI techniques for disaster management. Most of these techniques use deep learning methods to automatically extract information from satellite imagery [START_REF] Rahnemoonfar | Floodnet: A high resolution aerial imagery dataset for post flood scene understanding[END_REF][START_REF] Gupta | Creating xbd: A dataset for assessing building damage from satellite imagery[END_REF][START_REF] Dejito | Mapping access to water and sanitation in colombia using publicly accessible satellite imagery, crowd-sourced geospatial information and randomforests[END_REF][START_REF] Xu | Building damage detection in satellite imagery using convolutional neural networks[END_REF]. These methods extract information like building and road damages, flood maps, population maps, and wildfire maps. It has been claimed that such derived information can help automate decision-making during a disaster for quick and better response. In the last three years of HADR workshops, there have been 13 papers proposing building damage assessment, 3 proposing flood mapping, and 6 proposing better ways of relief management. The context of this paper is to check the alignment of existing methods with the actual need of disaster victims. We analyze these methods' disaster time usability with domain experts at US Air Force Research Laboratory (AFRL). We find that the current research is directed toward obtaining better metrics that satisfy machine learning criteria and are not very useful at achieving the core beliefs mentioned above.

2 Sphere AI Guidelines

The Sphere Project was started in 1997 by a group of humanitarian non-governmental organizations and the Red Cross and Red Crescent Movement. They aim to improve the quality of humanitarian responses and be accountable for actions. The core beliefs of Sphere includes : 1) the right to life with dignity, 2) the right to humanitarian assistance, and 3) the right to protection and security. Humanitarian standards are developed by humanitarian practitioners with specific areas of expertise and are formulated based on evidence, experience, and learning. The standards are developed by consensus and reflect accumulated best practices at a global level. They are revised regularly to incorporate developments across the sector. Humanitarian standards are valuable for practitioners to plan, implement, and evaluate humanitarian responses. They provide a solid basis to advocate for the rights of crisis-affected communities. By applying them in preparedness and response operations, humanitarian agencies and practitioners commit to quality and make themselves accountable to the populations they serve.

The Sphere handbook sets principles and minimum humanitarian standards in four technical areas of humanitarian response: 1) Water supply, sanitation, and hygiene promotion (WASH), 2) Food security and nutrition, 3) Shelter and settlement, and 4) Health. It also discusses the vulnerabilities, gaps, and difficulties from various perspectives that need to be considered and taken care of for an effective humanitarian response. The presented standards and guidance in the Sphere Handbook are supported by research and evidence that reflect 20 years of field testing by humanitarians worldwide. The principal users of the Sphere Handbook are practitioners involved in planning, managing, and implementing a humanitarian response. They identify the key indicators in all the four technical areas necessary for providing minimum standards in case of humanitarian response. For example, in the case of WASH, key indicators include "Percentage of water quality tests meeting minimum water quality standards" and "Percentage of toilets reported as safe by women and girls." These key indicators are essential in pre-and poster disaster scenarios. Also, Sphere identifies the needs and requirements of the people at both individual and community levels. Aligning the guidelines in the Sphere handbook with Satellite imagery, AI can help identify water points, shelter points, waste disposal units, disease breeding sites, difficult-to-access areas, and nearby health centers. Engaging and following such guidelines will help provide more pragmatic and useful solutions.

In the AI community, technologists and researchers are moving towards developing a more socially responsible AI, which is of utmost importance when deploying the systems for areas as critical and vulnerable as HADR. The social responsibility of AI includes efforts devoted to addressing technical and societal issues. The four fundamental responsibilities of social AI [START_REF] Cheng | Socially responsible ai algorithms: Issues, purposes, and challenges[END_REF] include 1) functional, 2) legal, 3) ethical, and 4) philanthropic. For the most part, AI practice considers the fundamental responsibilities of AI systems to be functional and legal, whereas ethical and philanthropic responsibilities are somewhat neglected in practice. The Sphere handbook can be an excellent starting point for AI practitioners to align their upcoming work with humanitarian response's ethical, legal, and practical basis. This will help in developing systems that are good AI Citizens. Sphere handbook also discusses Humanitarian responses in different contexts and under various circumstances, such as the differences across populations and diversity among individuals, operational and logistical realities that will affect how and what kind of humanitarian response is delivered. Considering these facts, it is also crucial for AI systems to keep up with their functional responsibilities and make more generalized and robust systems.

Current Status of Geospatial AI in Disaster Relief

Artificial Intelligence (AI), specifically machine vision and deep learning, allow us to interpret imagery at scale in near-real-time [START_REF] Daudt | Urban change detection for multispectral earth observation using convolutional neural networks[END_REF][START_REF] Papadomanolaki | Detecting urban changes with recurrent neural networks from multitemporal sentinel-2 data[END_REF][START_REF] Verma | Qfabric: Multi-task change detection dataset[END_REF]. Using AI centered tools like GeoEngine [START_REF] Verma | Geoengine: A platform for production-ready geospatial research[END_REF][START_REF] Shin | Europa: Increasing accessibility of geospatial datasets[END_REF], we can further increase humanitarian capacity building by simplifying access to ground truth data. This is particularly important after a disaster when compromised physical infrastructure could threaten ground responders and limit access to lines of communication. Remotely sensed data can provide immediate data to triage and plan. It can also provide pre-disaster baseline data. Further, autonomous analysis techniques can introduce novel spatial metrics for a nuanced understanding of a given region's condition. table shows what type of machine learning problems existing geospatial datasets and methods try to solve in disaster relief and management goals. It also shows if they align with the guidelines in The Sphere Handbook. Following disaster types have been considered: E arthquake/Tsunami, F lood, L andslide, V olcanic Eruption, and W ildfire.

Research community started with the intention to detect after-effects of disasters like damage assessment and flood mapping. The goal has been to automatically analyze a large swath of area in the least possible time with very little human intervention. Datasets like OSCD [START_REF] Daudt | Urban change detection for multispectral earth observation using convolutional neural networks[END_REF][START_REF] Papadomanolaki | Detecting urban changes with recurrent neural networks from multitemporal sentinel-2 data[END_REF] and QFabric [START_REF] Verma | Qfabric: Multi-task change detection dataset[END_REF] enable training change detection models to detect construction and destruction of man made structures. Datasets like xView2 [START_REF] Gupta | Creating xbd: A dataset for assessing building damage from satellite imagery[END_REF] and FloodNet [START_REF] Rahnemoonfar | Floodnet: A high resolution aerial imagery dataset for post flood scene understanding[END_REF] enable training large deep learning models for building damage detection and classification. Several methods have been proposed on these datasets that aim to improve the F1-score [START_REF] Shen | Cross-directional feature fusion network for building damage assessment from satellite imagery[END_REF][START_REF] Khvedchenya | Fully convolutional siamese neural networks for buildings damage assessment from satellite images[END_REF][START_REF] Xia | Building damage mapping with self-positive unlabeled learning[END_REF] or solve some niche problems [START_REF] Benson | Assessing out-of-domain generalization for robust building damage detection[END_REF][START_REF] Shin | Characterizing human explanation strategies to inform the design of explainable ai for building damage assessment[END_REF][START_REF] Kuzin | Disaster mapping from satellites: damage detection with crowdsourced point labels[END_REF].

Benson et al. [START_REF] Benson | Assessing out-of-domain generalization for robust building damage detection[END_REF] argues that models trained on datasets like xView2 perform worse when new images come, and they try to solve the problem of concept drift. This is a critical problem to understand and resolve. This has also been discussed in the Sphere handbook as the problem where a disaster occurs first time in a new area for which existing models do not exist or are bound to fail. Similarly, [START_REF] Kuzin | Disaster mapping from satellites: damage detection with crowdsourced point labels[END_REF] proposed using human-in-the-loop to adapt existing damage detection models to a new area by using sparse inputs in the form of point labels by experts available on the ground. Decision makers on the ground during disaster relief and management are key people, and as much we would like to use machine learning based methods to make things fast, their eyes on such ML outputs are quite important. Explainable AI comes to the rescue for such requirements as shown by [START_REF] Shin | Characterizing human explanation strategies to inform the design of explainable ai for building damage assessment[END_REF]. Another important factor the Sphere handbook mentions is that different places have different rules on data acquisition and decimation. Furthermore, high-quality data is often unavailable and expensive to acquire, especially in the case of aerial images. To solve such problems often, small unmanned aerial vehicles (sUAV) mounted with inexpensive handheld cameras like GoPro are used [START_REF] Franceschini | Damage estimation and localization from sparse aerial imagery[END_REF]. One crucial technical area of humanitarian response is water supply and hygiene promotion (WASH). [START_REF] Dejito | Mapping access to water and sanitation in colombia using publicly accessible satellite imagery, crowd-sourced geospatial information and randomforests[END_REF] proposes leveraging satellite images and public data to map water availability and population dependency during the Venezuelan migration crisis.

After analysing all the current papers specially one published in last three years of HADR, we found that most methods are focused on detecting building damage. Only Tingzon et al. [START_REF] Dejito | Mapping access to water and sanitation in colombia using publicly accessible satellite imagery, crowd-sourced geospatial information and randomforests[END_REF] work on WASH preparedness using satellite imagery. Building damage assessment methods provide very macro analysis of disaster destruction. They do not provide intervention-level insights aligned with the need of policymakers and development organizations. We propose that Geospatial AI capabilities be extended/mapped to align critical indicators in the Sphere handbook.

Case Study of WASH

In this study we aim to calculate key indicators for water safety during flooding. We use guidelines mentioned in WASH technical area in the Sphere handbook. For this we collect SkySat and Pleiades satellite images (50cm resolution) of flooded regions in Pakistan between July -September 2022. To date, more than 1.14 million houses have been damaged and over 765,000 houses have been destroyed across the country, while over 1,500 deaths and more than 12,800 injuries were recorded since mid-June. Figure 1 shows satellite images captured over Sehwan Sharif airport before and after the floods. Before and after images for two more cities, Kairpur and Qambar were also collected.

Key Indicators for Water Safety

The Sphere handbook sets certain indicators for water safety under the technical area of WASH. We aim to derive these indicators from satellite imagery using existing deep learning methods. Following are the key indicators for water safety:

• Average volume of water used for drinking and domestic hygiene per household. Minimum of 15 liters per person per day and determine quantity based on context and phase of the response.

• Maximum number of people using the water-based facility. 250 people per tap, 500 people per hand pump, 400 people per open hand well, 100 people per laundry facility, and 50 people per bathing facility.

• Percentage of household income used to buy water for drinking and domestic hygiene: target 5 percent or less.

• Percentage of targeted households who know where and when they will next get their water.

• Distance from any household to the nearest water point: <500 meters.

• Queuing time at water sources: <30 minutes.

• Percentage of communal water distribution points free of standing water.

• Percentage of water systems/facilities that have functional and accountable management system in place.

Flood Detection

The first step in the above key indicators is identifying houses, schools, drinking water bodies, and flooded areas in satellite images. We use BiDate UNet [START_REF] Daudt | Urban change detection for multispectral earth observation using convolutional neural networks[END_REF][START_REF] Papadomanolaki | Detecting urban changes with recurrent neural networks from multitemporal sentinel-2 data[END_REF] trained on xView2 dataset [START_REF] Gupta | Creating xbd: A dataset for assessing building damage from satellite imagery[END_REF] to detect buildings and classify there damage levels. xView2 dataset only detects buildings and their damage levels. We also train flood foundation network available as SpaceNet-8 baseline2 to get flood maps. SpaceNet-8 contains images from New Orleans, Louisiana, following Hurricane Ida in August 2021 and Dernau, Germany, during the June 2021 floods across Western Europe. The trained networks turned out to be heavily biased toward the urban planning of these cities and failed to give results for the small towns and villages of Pakistan.

Other Essential Information

Population maps were derived for Shewan, Qambar, and Khairpur from Data for Good3 . To obtain existing locations were safe water can be obtained we look into mWater 4 and UN Wash5 projects. UN Wash map only shows country-level metrics around the availability of safe water during normal times. mWater shows a more up-to-date map of safe water availability but is available for only 0.1% area of Pakistan. Figure 2 shows information derived using the trained model and Data for Good population maps. We manually identify government buildings to obtain vital areas where safe water can be available during a flood. Qambar and Kharipur have 5 and 2 government buildings within the observed area. We use all this information to calculate key indicators. Within the satellite image captured for Qambar (6.04 km-square) and Khairpur (1.68 km-square), there are approximately 61187 and 11664 people, respectively. Assuming that all the government buildings have safe tap water available, 12237 and 5832 people depend on them. This is way above then the set of 250 people per tap water. The average distance from the household is 350 meters and 410 meters for Qambar and Khairpur, respectively.

Conclusion

This paper presents the gap the Sphere handbook's international guidelines and current geospatial AI HADR research. We briefly describe the Sphere handbook and how it can help us design better geospatial AI solutions. Using existing methods and derived information to calculate water safety (WASH) key indicators for two cities in Pakistan for 2022 floods. We hope this study will steer the HADR research to align with international guidelines leading to highly focused solutions for individuals affected by disasters.

Figure 1 :

 1 Figure 1: Sehwan Sharif Airport and nearby villages captured on 2022-01-27 and after the flood on 2022-09-24.

Figure 2 :

 2 Figure 2: Derived information from satellite images. The left column shows Khairpur, and the right shows Qambar. The top row shows detected buildings, and the bottom row shows population density per 30m × 30m grid.

Table 1 :

 1 Table 1 lists out some very famous datasets and methods proposed in last few years that aim at disaster relief and management. This

	Method/Dataset	Disasters	ML Task	Alignment
	xView2 (xBD) [2]	E F L V W	Detection	No
	CDF [11]	E F L V W	Detection	No
	Benosn et al., 2020 [12]	E F L V W	Concept Drift	Yes
	Khvedchenya et al., 2021 [13] E F L V W	Detection	No
	Xia et al., 2021 [14]	E F L V W	Detection	No
	Shin et al., 2021 [15]	E F L V W	Explainability	Yes
	Kuzin et al., 2021 [16]	E F L V W	Human-in-the-loop	Yes
	FloodNet [1]	F	Detection	No
	SpaceNet-8 [17]	F	Detection	No
	Franceschini et al., 2021 [18]	F	Acquisition	Yes
	Ochoa et al., 2021 [19]	E F	Detection	No
	Seo et al., 2019 [20]	E	Data Imbalance	No
	BRAILS [21]	E	Soft-Story Buildings	No
	Xu et al., 2019 [4]	E	Detection	No
	Boin et al., 2020 [22]	E	Data Imbalance	No
	Lee et al., 2020 [23]	E W	Detection	No
	FireNet [24]	W	Fire Perimeter	No
	Huot et al., 2020 [25]	W	Wildfire Prediction	No

https://spherestandards.org/handbook-2018/ Preprint. Under review.

https://github.com/SpaceNetChallenge/SpaceNet8

https://dataforgood.facebook.com/dfg/tools/high-resolution-population-density-maps

https://bit.ly/3rrMecs

https://washdata.org/data/household