**4. Conditions imposed over the homogeneous Fredholm equations**

In accordance with the theory of homogeneous Fredholm integral equations [1, 2, 13], the first Fredholm minor is a two point function, like a Green function, which must comply with an integral equation:

$$\begin{aligned} \odot \ell^{\mathfrak{m}}(\mathbf{r}, \mathbf{r}\_0; \boldsymbol{\alpha}) &= \eta(\boldsymbol{\alpha}) \Delta(\boldsymbol{\eta}, \boldsymbol{\alpha}) \\\\ &+ \eta(\boldsymbol{\alpha}) \int\_0^\infty \mathbf{K}\_n^{m(\circ)}(\boldsymbol{\alpha}; \mathbf{r}, \mathbf{s}) \odot \ell^n(\mathbf{s}, \mathbf{r}\_0; \boldsymbol{\alpha}) d\boldsymbol{s} \end{aligned} \tag{7}$$

Two other conditions must be satisfied: The first is that Fredholm determinant is zero

$$
\Delta(\eta, \alpha) = 0 \tag{8}
$$

The second that the Fredholm eigenvalue equals to one:

$$\eta(a) = \mathbf{1} \tag{9}$$

But thanks to our second order approximation Eq. (2) we can show that other interesting conditions are satisfied, for example if we define some particular functions (and operators):

$$\Psi(\mathbf{r};\alpha) \equiv \odot \ell^{\mathfrak{m}}(\mathbf{r},\mathbf{r}\_{0};\alpha) - \Delta(\eta,\alpha)u^{\mathfrak{m}}(\mathbf{r};\alpha) \tag{10}$$

And also

$$\begin{split} \Psi^{(\circ)}(\mathbf{r};\boldsymbol{\omega}) & \equiv \Delta(\boldsymbol{\eta},\boldsymbol{\omega}) \Big[ \boldsymbol{\eta}(\boldsymbol{\omega}) - \boldsymbol{\mu}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega}) \Big] \\ &+ \Delta(\boldsymbol{\eta},\boldsymbol{\alpha}) [\boldsymbol{\eta}(\boldsymbol{\alpha}) - \boldsymbol{\nu}(\boldsymbol{\alpha})] \int \mathbf{K}\_{\boldsymbol{\eta}}^{m(\circ)}(\boldsymbol{\omega};\mathbf{r},\mathbf{r}') \boldsymbol{\mu}^{n}(\mathbf{r}';\boldsymbol{\omega}) d\boldsymbol{r}' \end{split} \tag{11}$$

The same operator (15) acting over a homogeneous equation looks like

þ Δð Þ*η* ½ � *η* � 0

*e* ð Þ¼ **r**; *ω*

*e* ð Þ¼ **r**;*ω λym*

Z **r**; *ω*; *um*ð Þ∘ ð Þ **r**;*ω*

� � (18)

*i δ*ð Þ **r** � **r***i* **K***m*ð Þ∘ *n* ð Þ *ω*; **r**, **r***i* (19)

!

*i δ*ð Þ **r** � **r***i* **K***m*ð Þ∘ *n* ð*ω*; **r**, **r***i*Þ

� (20)

*e* ð Þ **r**; *ω* (21)

*e* **r**0 ð Þ ;*ω dr*0

(16)

*e* ð Þ **r**;*ω* (17)

**K***m*ð Þ∘ *n ω*; **r**, **r**0 ð Þ*yn*

As we can see the effect of the Zap operator is to kill or eliminate the inhomogeneous term when applied to a resonant state. But this seems very artificial because we are giving indeed two parts for the complete rule. However we can build

On this section, we define the so named Zap projection operators (ZPO) which enable us to project a complex broadcasting system over a reduced resonant simplest one. The Zap operators acts over Fourier transforms [14, 15] related to integral

In order to get a display of the properties of this operator we propose a specific

Then, by applying the projection operator to Eq. (19) we have (remember that

½Δð Þ*η η* �X

**K***m*ð Þ∘ *n ω*; **r**, **r**0 ð Þ*un* **r**0 ð Þ ; *ω dr*0

h i*um*ð Þ¼ **r**; *ω ym*

*p*

*i*¼1 *αn*

Suppose that we have *p* punctual sources that can be represented in the

*p*

*i*¼1 *αn*

*e* ð Þ� **r**;*ω* Z½ � **r**; *ω*; 0 *ym*

þ *η* ∞ð

*e* ð Þ� **r**; *ω* Z½ � **r**;*ω*; 0 *ym*

Now, we define Zap projection operators in the next section.

Then, based on (15) and (17), we define de following operator:

� � ¼ lim*η*!1

**6. The zap projection operators and their properties**

*P* **r**;*ω*; *um*ð Þ∘ ð Þ **r**; *ω*

set of discrete antennas in the next example:

h i*um*ð Þ¼ **r**;*ω* lim*η*!1

when *η* ¼ 1 thus Δ ¼ 0):

*P* **r**;*ω*; *um*ð Þ∘ ð Þ **r**; *ω*

**5**

inhomogeneous term of the Fredholm equation like:

*um*ð Þ∘ ð Þ¼ **r**;*ω* X

þ*η* ∞ð

0

*P* **r**;*ω*; *um*ð Þ∘ ð Þ **r**; *ω*

Now, because *η* ¼ 1 implies Δ ¼ 0, and because also *ν* ¼ *λ* ¼ 1

0

Z*ym*

*On the Zap Integral Operators over Fourier Transforms DOI: http://dx.doi.org/10.5772/intechopen.94573*

Z*ym*

projection operators that can make the work we need.

That is

operators.

We can see that the first Fredholm minor must satisfy through Ψ the inhomogeneous equation

$$\Psi(\mathbf{r};\boldsymbol{\alpha}) \equiv \Psi^{(\circ)}(\mathbf{r};\boldsymbol{\alpha}) + \eta(\boldsymbol{\alpha}) \int\_{0}^{\boldsymbol{\alpha}} \mathbf{K}\_{n}^{m(\circ)}(\boldsymbol{\alpha};\mathbf{r},\mathbf{t}) \Psi(\mathbf{t};\boldsymbol{\alpha}) d\mathbf{t} \tag{12}$$

In order to write Eq. (2) in terms of the solutions of Eq. (1), we can define the operator:

$$\Theta^{\mathfrak{m}}(\mathbf{r};\boldsymbol{\omega}) \equiv \nu^2(\boldsymbol{\alpha})\boldsymbol{\varepsilon}(\boldsymbol{\alpha}) \left[ \mathbf{K}\_{\boldsymbol{\alpha}}^{\mathfrak{m}(\boldsymbol{\ast})}(\boldsymbol{\alpha};\mathbf{r},\mathbf{r}') \right] \bigg|\_{\mbox{\tiny{\tiny}}\nolimits \mbox{\tiny{\tiny}}\nolimits \mbox{\tiny{\scriptsize{\textsc{\tiny}}}}{\mbox{\scriptsize{\textsc{\scriptsize{\textsc{\tiny}}}}}}(\boldsymbol{\alpha};\mathbf{r}',\mathbf{r}') \mathrm{P}^{\mathfrak{m}}(\mathbf{r}';\boldsymbol{\alpha}) dr' dr' \tag{13}$$

In Eq. (13) the function P*m*ð Þ **r**; *ω* is an arbitrary negative exponential regulator.

Near a resonance the two small parameters *ν ω*ð Þ and *ε ω*ð Þ makes Θ*m*ð Þ **r**; *ω* lesser than a second order term, so can be neglected. Far of a resonance this later function sketches the behavior of the simultaneous existence of the resonant and nonresonant solutions because in terms of Θ*m*ð Þ **r**; *ω* the conventional waves satisfy the inhomogeneous equation:

$$\begin{split} u^{m}(\mathbf{r};\boldsymbol{\alpha}) &= u^{m(\circ)}(\mathbf{r};\boldsymbol{\alpha}) + \Theta^{m}(\mathbf{r};\boldsymbol{\alpha}) \\ &+ \nu(\boldsymbol{\alpha}) \int\_{0}^{\boldsymbol{\alpha}} \mathbf{K}\_{n}^{m(\circ)}(\boldsymbol{\alpha};\mathbf{r},\mathbf{r}') u^{n}(\mathbf{r}';\boldsymbol{\alpha}) d\boldsymbol{r}' \end{split} \tag{14}$$

### **5. Defining a new class of integral operators**

As we said in Section **1**, hidden in the structure of Eq. (2) there are some integral operators which allow the simultaneous existence of solutions with extremely different boundary conditions. So, let us define the Zap operators by the rules:

$$\begin{aligned} \mathbf{Z}u^{m}(\mathbf{r};\boldsymbol{\alpha}) & \equiv \mathbf{Z} \Big[ \mathbf{r};\boldsymbol{\alpha};u^{m(\circ)}(\mathbf{r};\boldsymbol{\alpha}) \Big] u^{m}(\mathbf{r};\boldsymbol{\alpha}) = \\ & + \Delta(\boldsymbol{\eta}) \Big[ \boldsymbol{\eta} - u^{m(\boldsymbol{\alpha})}(\mathbf{r};\boldsymbol{\alpha}) \Big] \\ & + \eta \int\_{0}^{\boldsymbol{\alpha}} \mathbf{K}\_{n}^{m(\circ)}(\boldsymbol{\alpha};\mathbf{r},\mathbf{r}') u^{n}(\mathbf{r}';\boldsymbol{\alpha}) d\boldsymbol{r}' \end{aligned} \tag{15}$$

That is, the Zap operator is associated to the integral Fredholm equation satisfied by the affected solution (*um*ð Þ **r**;*ω* or *ym e* ð Þ **r**; *ω* ), from which takes the source term and the free kernel.

The same operator (15) acting over a homogeneous equation looks like

$$\begin{aligned} \mathbf{Z} \mathbf{y}\_{\epsilon}^{m}(\mathbf{r}; \boldsymbol{\alpha}) & \equiv \mathbf{Z}[\mathbf{r}; \boldsymbol{\alpha}; \mathbf{0}] \mathbf{y}\_{\epsilon}^{m}(\mathbf{r}; \boldsymbol{\alpha}) = \\ &+ \Delta(\boldsymbol{\eta})[\boldsymbol{\eta} - \mathbf{0}] \\ &+ \eta \int\_{0}^{\boldsymbol{\alpha}} \mathbf{K}\_{n}^{m(\boldsymbol{\epsilon})}(\boldsymbol{\alpha}; \mathbf{r}, \mathbf{r}') \mathbf{y}\_{\epsilon}^{n}(\mathbf{r}'; \boldsymbol{\alpha}) d\mathbf{r}' \end{aligned} \tag{16}$$

That is

And also

geneous equation

Θ*m*ð Þ� **r**;*ω ν*2

inhomogeneous equation:

ð Þ *ω ε ω*ð Þ

ð ∞

**K***m*ð Þ∘ *n ω*; **r**, **r**0 ð Þ

*um*ð Þ¼ **r**;*ω um*ð Þ∘ ð Þþ **r**;*ω* Θ*m*ð Þ **r**;*ω*

∞ð

0

operators which allow the simultaneous existence of solutions with extremely different boundary conditions. So, let us define the Zap operators by the rules:

Z*um*ð Þ� **r**;*ω* Z **r**;*ω*; *um*ð Þ∘ ð Þ **r**; *ω*

þ *η* ∞ð

0

As we said in Section **1**, hidden in the structure of Eq. (2) there are some integral

h i

þ Δð Þ*η η* � *um*ð Þ o ð Þ **r**; *ω* h i

That is, the Zap operator is associated to the integral Fredholm equation satisfied

**K***m*ð Þ∘ *n ω*; **r**, **r**0 ð Þ*un* **r**0 ð Þ ;*ω dr*0

þ *ν ω*ð Þ

**5. Defining a new class of integral operators**

by the affected solution (*um*ð Þ **r**;*ω* or *ym*

and the free kernel.

**4**

0

operator:

Ψð Þ∘ ð Þ� **r**;*ω* Δð Þ *η*, *ω η ω*ð Þ� *um*ð Þ∘ ð**r**;*ω*Þ

þ Δð Þ *η*,*ω* ½ � *η ω*ð Þ� *ν ω*ð Þ

Ψð Þ� **r**;*ω* Ψð Þ∘ ð Þþ **r**;*ω η ω*ð Þ

h i

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

∞ð

**K***m*ð Þ∘ *n ω*; **r**, **r**0 ð Þ*un* **r**0 ð Þ ; *ω dr*0 (11)

**K***m*ð Þ∘ *n* ð Þ *ω*; **r**,**t** Ψð Þ **t**; *ω dt* (12)

*dr*0 (13)

(15)

, **r**00 ð ÞP*m* **r**00 ð Þ ; *ω dr*00

**K***m*ð Þ∘ *n ω*; **r**, **r**0 ð Þ*un* **r**0 ð Þ ; *ω dr*0 (14)

*um*ð Þ¼ **r**; *ω*

*e* ð Þ **r**; *ω* ), from which takes the source term

0

We can see that the first Fredholm minor must satisfy through Ψ the inhomo-

∞ð

0

In order to write Eq. (2) in terms of the solutions of Eq. (1), we can define the

∞ð

**K***n*ð Þ∘ *l ω*; **r**0

0

In Eq. (13) the function P*m*ð Þ **r**; *ω* is an arbitrary negative exponential regulator. Near a resonance the two small parameters *ν ω*ð Þ and *ε ω*ð Þ makes Θ*m*ð Þ **r**; *ω* lesser than a second order term, so can be neglected. Far of a resonance this later function sketches the behavior of the simultaneous existence of the resonant and nonresonant solutions because in terms of Θ*m*ð Þ **r**; *ω* the conventional waves satisfy the

$$\mathbf{Z}\mathbf{y}\_{\varepsilon}^{m}(\mathbf{r};\alpha) \equiv \mathbf{Z}[\mathbf{r};\alpha;\mathbf{0}]\mathbf{y}\_{\varepsilon}^{m}(\mathbf{r};\alpha) = \lambda \mathbf{y}\_{\varepsilon}^{m}(\mathbf{r};\alpha) \tag{17}$$

As we can see the effect of the Zap operator is to kill or eliminate the inhomogeneous term when applied to a resonant state. But this seems very artificial because we are giving indeed two parts for the complete rule. However we can build projection operators that can make the work we need.

Now, we define Zap projection operators in the next section.

### **6. The zap projection operators and their properties**

On this section, we define the so named Zap projection operators (ZPO) which enable us to project a complex broadcasting system over a reduced resonant simplest one. The Zap operators acts over Fourier transforms [14, 15] related to integral operators.

Then, based on (15) and (17), we define de following operator:

$$\mathbb{Z}\_P\left(\mathbf{r}; \boldsymbol{\alpha}; \boldsymbol{u}^{m(\circ)}(\mathbf{r}; \boldsymbol{\alpha})\right) = \lim\_{\eta \to 1} \mathbb{Z}\left(\mathbf{r}; \boldsymbol{\alpha}; \boldsymbol{u}^{m(\circ)}(\mathbf{r}; \boldsymbol{\alpha})\right) \tag{18}$$

In order to get a display of the properties of this operator we propose a specific set of discrete antennas in the next example:

Suppose that we have *p* punctual sources that can be represented in the inhomogeneous term of the Fredholm equation like:

$$u^{m(\circ)}(\mathbf{r};w) = \sum\_{i=1}^{p} a\_i^n \delta(\mathbf{r} - \mathbf{r}\_i) \mathbf{K}\_n^{m(\circ)}(w; \mathbf{r}, \mathbf{r}\_i) \tag{19}$$

Then, by applying the projection operator to Eq. (19) we have (remember that when *η* ¼ 1 thus Δ ¼ 0):

$$\mathbb{Z}\_P\left[\mathbf{r};\,\boldsymbol{\alpha};\,\boldsymbol{u}^{m(\circ)}(\mathbf{r};\,\boldsymbol{\alpha})\right]\boldsymbol{u}^m(\mathbf{r};\,\boldsymbol{\omega})=\lim\_{\eta\to 1}\left[\Delta(\eta)\left(\eta-\sum\_{i=1}^p\alpha\_i^n\delta(\mathbf{r}-\mathbf{r}\_i)\mathbf{K}\_n^{m(\circ)}(\boldsymbol{\alpha};\,\mathbf{r},\mathbf{r}\_i)\right)\right]$$

$$+\eta\int\_0^\infty\mathbf{K}\_n^{m(\circ)}(\boldsymbol{\alpha};\,\mathbf{r},\mathbf{r}')\boldsymbol{u}^n(\mathbf{r}';\boldsymbol{\alpha})d\boldsymbol{r}'\right]\tag{20}$$

Now, because *η* ¼ 1 implies Δ ¼ 0, and because also *ν* ¼ *λ* ¼ 1

$$\mathbb{Z}\_P\Big[\mathbf{r};\,\boldsymbol{\alpha};\,\boldsymbol{u}^{m(\ast)}(\mathbf{r};\,\boldsymbol{\alpha})\Big]\boldsymbol{u}^m(\mathbf{r};\,\boldsymbol{\alpha})=\mathcal{Y}\_\epsilon^m(\mathbf{r};\,\boldsymbol{\alpha})\tag{21}$$

In the last step we have used the fact that the solution of the remaining homogeneous equation is denoted by *ym e* ð Þ **r**;*ω* .

Eq. (21) says that if we take a blend of regular and resonant solutions we have:

$$\mathbb{Z}\_P\left[\mathbf{r};\,\alpha;\,\mu^{m(\ast)}(\mathbf{r};\,\alpha)\right](\boldsymbol{u}^m(\mathbf{r};\,\alpha)+\boldsymbol{\mathcal{y}}\_\epsilon^m(\mathbf{r};\,\alpha))=\mathbf{2}\mathbf{y}\_\epsilon^m(\mathbf{r};\,\alpha)\tag{22}$$

In Eq. (25) *A*ð Þ**r** is the interaction that in the general case may contain a non-

*u*2ð Þ **r**;*ω* � � � **K**ð Þ∘ ð Þ *ω g*1ð Þ *ω*

cos *ω* � *ωp* � �*d*

*ω* � *ωp* � �*d*

*αiδ*ð Þ **r** � **r***i* **K**ð Þ∘ ð Þ *ω* **e***i* (28)

**K**ð Þ∘ *ω*; **r**, **r**0 ð Þ*u* **r**0 ð Þ ,*ω dr*0 (31)

1

CCCCA

*u*ð Þ**r** (33)

sin *ω* � *ωp* � �*d*

*ω* � *ωp* � �*d*

1

*u*ð Þ∘ ð Þ¼ **r**; *ω α*1*δ*ð Þ **r** � **r**1 **K**ð Þ∘ ð Þ *ω* **e**1 þ *α*2*δ*ð Þ **r** � **r**2 **K**ð Þ∘ ð Þ *ω* **e**2 (30)

*g*2ð Þ *ω*

1

CCCA

� � (29)

(27)

(32)

� �*u*ð Þ**r** (26)

local potential, but not in our example.

*On the Zap Integral Operators over Fourier Transforms DOI: http://dx.doi.org/10.5772/intechopen.94573*

In Eq. (26) *u*ð Þ**r** is a scalar function. And then, the kernel may be

**K**ð Þ∘ ð Þ¼ *ω*

11 **K**ð Þ∘ 12

*i*

*u*ð Þ∘ ð Þ¼ **r**;*ω* X

**e**1 ¼ 1 0

The conventional waves satisfy the scalar form of Eq. (1)

*u*ð Þ¼ **r**, *ω u*ð Þ∘ ð Þ **r**, *ω*

þ *λ ω*ð Þ

∞ð

0

*u*ð Þ¼ **r**, *ω u*ð Þ∘ ð Þ **r**, *ω*

Where the form of *u*ð Þ **r**,*ω* is unknown but possibly be sketched as

0

BBBB@

*u*ð Þ¼ **r**,*ω*

þ *λ ω*ð Þ**K**ð Þ∘ ð Þ *ω u*ð Þ **r**, *ω*

sin *ω* � *ωp* � �*d*

*ω* � *ωp* � �*d* sin *ω* � *ωp* þ *β* � �*d ω* � *ωp* þ *β* � �*d*

Now we can apply the projection operator *P* **r**; *ω*; *u*ð Þ∘ ð Þ **r**;*ω* � � to Eq. (32) and

0

BBB@

! *u*1ð Þ **r**; *ω*

sin *ω* � *ωp* � �*d*

*ω* � *ωp* � �*d* �*i*

cos *ω* � *ωp* � �*d*

*ω* � *ωp* � �*d*

2

*i*¼1

� � and**e**2 ¼ 0

**K**ð Þ∘ 21 **K**ð Þ∘ 22

**K**ð Þ∘ ð Þ *ω u*ð Þ� **r**; *ω* **K**ð Þ∘

So Eq. (19) takes the form:

Or in accordance with Eq. (25)

Where

That is

obtain

**7**

So taken into account from Eqs. (18) until (22), we see that we have projected the original problem into a resonant one.

In analogy with *P* we can define a projector over their complement: Let us define the complementary Zap projection operator as

$$\begin{split} \mathbb{Z}\_{Q}\left(\mathbf{r};\boldsymbol{\omega};\boldsymbol{u}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega})\right)\boldsymbol{u}^{m}(\mathbf{r};\boldsymbol{\omega}) & \equiv \lim\_{\eta \to 1} \boldsymbol{Z}^{C}\Big(\mathbf{r};\boldsymbol{\omega};\boldsymbol{u}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega})\Big) \\ & \equiv \lim\_{\eta \to 1} \left\{ (\Delta(\eta)\eta + \nu) \int\_{0}^{\infty} \mathbf{K}\_{\boldsymbol{u}}^{m(\circ)}(\boldsymbol{\omega};\mathbf{r},\mathbf{r}')\boldsymbol{u}^{n}(\mathbf{r}';\boldsymbol{\omega}) d\boldsymbol{r}' \right. \\ & \left. + \eta \boldsymbol{u}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega})\right\} = \boldsymbol{u}^{m}(\mathbf{r};\boldsymbol{\omega}) \end{split} \tag{23}$$

Even we apply *Q* to a resonant state:

$$\begin{split} \mathbb{Z}\_{\mathcal{Q}}\left(\mathbf{r};\boldsymbol{\omega};\boldsymbol{u}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega})\right)\boldsymbol{\uprho}\_{\epsilon}^{m}(\mathbf{r};\boldsymbol{\omega}) &\equiv \lim\_{\eta\to 1} \boldsymbol{Z}^{C}\Big(\mathbf{r};\boldsymbol{\omega};\boldsymbol{u}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega})\Big) \\ &= \lim\_{\eta\to 1} \left\{ (\boldsymbol{\Delta}(\eta)\boldsymbol{\eta}+\boldsymbol{\nu}) \int\_{0}^{\infty} \mathbf{K}\_{n}^{m(\circ)}(\boldsymbol{\omega};\mathbf{r},\mathbf{r}')\boldsymbol{\uprho}\_{\epsilon}^{n}(\mathbf{r}';\boldsymbol{\omega})d\boldsymbol{r}' \right. \\ &\left. + \boldsymbol{\eta}\boldsymbol{u}^{m(\circ)}(\mathbf{r};\boldsymbol{\omega})\right\} = \boldsymbol{u}^{m}(\mathbf{r};\boldsymbol{\omega}) \end{split} \tag{24}$$

This is because the name of the solution of the remaining inhomogeneous equation is precisely *um*ð Þ **r**; *ω* .

### **7. An academic example for conventional traveling waves**

In order to convince us of the utility of the *P* and *Q* operators we remember that in all of our developments the kernel always is **K***m*ð Þ∘ *n* that only contains the free Green function **G***t*ð Þ∘ *n* ð Þ *ω*; **r**, **s .** But then, there is no difference between the kernels of the integral equations when are referred to conventional traveling waves or to evanescent or resonant waves. This last statement allows describing in an algebraic mode the application of the Zap projection operators. In this manner we can fix our kernel in accordance with a previous example that we have presented in some place as the matrix (27).

For the case of only two source points and omitting the three components of the field lifting only one, this matrix can be for example:

But first remember that

$$\int\_{V} A(\mathbf{r}')G\_{\boldsymbol{\alpha}}^{(\circ)}(\mathbf{r};\mathbf{r}')u(\mathbf{r}';\boldsymbol{\alpha})dV' \equiv \mathbf{K}^{(\circ)}(\boldsymbol{\alpha})u(\mathbf{r};\boldsymbol{\alpha})\tag{25}$$

In Eq. (25) *A*ð Þ**r** is the interaction that in the general case may contain a nonlocal potential, but not in our example.

$$\mathbf{K}^{(\circ)}(w)u(\mathbf{r};w) \equiv \begin{pmatrix} \mathbf{K}\_{11}^{(\circ)} & \mathbf{K}\_{12}^{(\circ)} \\ \mathbf{K}\_{21}^{(\circ)} & \mathbf{K}\_{22}^{(\circ)} \end{pmatrix} \begin{pmatrix} u\_1(\mathbf{r};w) \\ u\_2(\mathbf{r};w) \end{pmatrix} \equiv \mathbf{K}^{(\circ)}(w) \begin{pmatrix} g\_1(w) \\ g\_2(w) \end{pmatrix} u(\mathbf{r}) \tag{26}$$

In Eq. (26) *u*ð Þ**r** is a scalar function. And then, the kernel may be

$$\mathbf{K}^{(\circ)}(\boldsymbol{\omega}) = \begin{pmatrix} \frac{\sin\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d}{\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d} & -i\frac{\cos\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d}{\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d} \\ i\frac{\cos\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d}{\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d} & \frac{\sin\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d}{\left(\boldsymbol{\omega} - \boldsymbol{\alpha}\_{p}\right)d} \end{pmatrix} \tag{27}$$

So Eq. (19) takes the form:

$$u^{(\circ)}(\mathbf{r};\boldsymbol{\alpha}) = \sum\_{i=1}^{2} a\_i \delta(\mathbf{r} - \mathbf{r}\_i) \mathbf{K}^{(\circ)}(\boldsymbol{\alpha}) \mathbf{e}\_i \tag{28}$$

Where

In the last step we have used the fact that the solution of the remaining homo-

Eq. (21) says that if we take a blend of regular and resonant solutions we have:

ð*um*ð Þþ **r**; *ω ym*

So taken into account from Eqs. (18) until (22), we see that we have projected

*e* ð ÞÞ ¼ **r**;*ω* 2*ym*

Z*C* **r**; *ω*; *um*ð Þ∘ ð Þ **r**;*ω* � �

Z*C* **r**; *ω*; *um*ð Þ∘ ð Þ **r**;*ω* � �

ð ∞

0

ð ∞

**K***m*ð Þ∘ *n ω*; **r**, **r**0 ð Þ*yn*

0

fð Þ Δð Þ*η η* þ *ν*

þ *ηum*ð Þ∘ ð Þg ¼ **r**;*ω um*ð Þ **r**; *ω*

fð Þ Δð Þ*η η* þ *ν*

þ *ηum*ð Þ∘ ð Þg ¼ **r**;*ω um*ð Þ **r**; *ω*

This is because the name of the solution of the remaining inhomogeneous equa-

In order to convince us of the utility of the *P* and *Q* operators we remember that in all of our developments the kernel always is **K***m*ð Þ∘ *n* that only contains the free

For the case of only two source points and omitting the three components of the

the integral equations when are referred to conventional traveling waves or to evanescent or resonant waves. This last statement allows describing in an algebraic mode the application of the Zap projection operators. In this manner we can fix our kernel in accordance with a previous example that we have presented in some place

*n* ð Þ *ω*; **r**, **s .** But then, there is no difference between the kernels of

*ω* **r**; **r**0 ð Þ*u* **r**0 ð Þ ; *ω dV*0 � **K**ð Þ∘ ð Þ *ω u*ð Þ **r**; *ω* (25)

*e* ð Þ **r**; *ω* (22)

**K***m*ð Þ∘ *n ω*; **r**, **r**0 ð Þ*un* **r**0 ð Þ ;*ω dr*0

(23)

(24)

*e* **r**0 ð Þ ; *ω dr*0

*e* ð Þ **r**;*ω* .

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

In analogy with *P* we can define a projector over their complement:

� lim*η*!1

Let us define the complementary Zap projection operator as

*um*ð Þ� **r**; *ω* lim*η*!1

*e* ð Þ� **r**; *ω* lim*η*!1

**7. An academic example for conventional traveling waves**

field lifting only one, this matrix can be for example:

*A* **r**0 ð Þ*G*ð Þ∘

¼ lim*η*!1

geneous equation is denoted by *ym*

*P* **r**; *ω*; *um*ð Þ∘ ð Þ **r**; *ω* h i

the original problem into a resonant one.

Even we apply *Q* to a resonant state:

*ym*

*Q* **r**;*ω*; *um*ð Þ∘ ð Þ **r**; *ω* � �

*Q* **r**; *ω*; *um*ð Þ∘ ð Þ **r**;*ω* � �

tion is precisely *um*ð Þ **r**; *ω* .

Green function **G***t*ð Þ∘

as the matrix (27).

**6**

But first remember that

ð

*V*

$$\mathbf{e}\_1 = \begin{pmatrix} \mathbf{1} \\ \mathbf{0} \end{pmatrix} \text{ and} \mathbf{e}\_2 = \begin{pmatrix} \mathbf{0} \\ \mathbf{1} \end{pmatrix} \tag{29}$$

That is

$$u^{(\ast)}(\mathbf{r};\boldsymbol{\alpha}) = a\_1 \delta(\mathbf{r} - \mathbf{r}\_1) \mathbf{K}^{(\ast)}(\boldsymbol{\alpha}) \mathbf{e}\_1 + a\_2 \delta(\mathbf{r} - \mathbf{r}\_2) \mathbf{K}^{(\ast)}(\boldsymbol{\alpha}) \mathbf{e}\_2 \tag{30}$$

The conventional waves satisfy the scalar form of Eq. (1)

$$\begin{aligned} u(\mathbf{r}, \omega) &= u^{(\circ)}(\mathbf{r}, \omega) \\ &+ \lambda(\omega) \int\_0^\infty \mathbf{K}^{(\circ)}(\boldsymbol{\omega}; \mathbf{r}, \mathbf{r}') u(\mathbf{r}', \omega) d\boldsymbol{r}' \end{aligned} \tag{31}$$

Or in accordance with Eq. (25)

$$\begin{aligned} u(\mathbf{r}, a) &= u^{(\circ)}(\mathbf{r}, a) \\ &+ \lambda(a) \mathbf{K}^{(\circ)}(a) u(\mathbf{r}, a) \end{aligned} \tag{32}$$

Where the form of *u*ð Þ **r**,*ω* is unknown but possibly be sketched as

$$u(\mathbf{r}, \omega) = \begin{pmatrix} \frac{\sin\left(\omega - \alpha\_p\right)d}{(\alpha - \alpha\_p)d} \\\\ \frac{\sin\left(\omega - \alpha\_p + \beta\right)d}{(\alpha - \alpha\_p + \beta)d} \end{pmatrix} u(\mathbf{r}) \tag{33}$$

Now we can apply the projection operator *P* **r**; *ω*; *u*ð Þ∘ ð Þ **r**;*ω* � � to Eq. (32) and obtain

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

$$\mathbb{Z}\_P\left[\mathbf{r};\boldsymbol{\omega};\boldsymbol{\upmu}^{(\circ)}(\mathbf{r};\boldsymbol{\upmu})\right]\boldsymbol{\upmu}(\mathbf{r};\boldsymbol{\upmu})=\lim\_{\eta\to 1}\left[\boldsymbol{\upLambda}(\eta)\left(\eta-\sum\_{i=1}^2a\_i\delta(\mathbf{r}-\mathbf{r}\_i)\mathbf{K}^{(\circ)}(\boldsymbol{\upmu})\mathbf{e}\_i\right)\right]$$

$$+\boldsymbol{\upmu}\mathbf{K}^{(\circ)}(\boldsymbol{\upmu})\boldsymbol{\upmu}(\mathbf{r};\boldsymbol{\upmu})\tag{34}$$

Then, by putting *η* ¼ 1 and Δ ¼ 0 finally

$$\mathbb{Z}\_P\left[\mathbf{r};\,\alpha;\,\mathfrak{u}^{(\ast)}(\mathbf{r};\,\alpha)\right]\mathfrak{u}(\mathbf{r};\,\alpha) = \mathfrak{y}\_\epsilon(\mathbf{r};\,\alpha) \tag{35}$$

So it is irrelevant the part of the problem concerning the two sources, it is only a problem about resonances. Our problem is now to find the resonant frequencies by taking **K**ð Þ∘ ð Þ *ω* and impose the conditions *η* ¼ 1 and Δ ¼ 0**.**

But, what is the real advantage of the *P* and *Q* operators?, the answer is that the Zap operator formalism may be viewed as a test for distinguish between an expression that cannot be transformed or yes, in whatever sense between the homogeneous and inhomogeneous equations under the rules established above; if not, we can ensure that some kind of irregular things are present. In case of the positive transformation, we have the confidence that both kinds of solutions can coexists, and then we can separate the solutions for convenience as if it was a problem of two steps: homogeneous and inhomogeneous.

Now the last condition over the Fredholm determinant is

$$
\Delta \begin{pmatrix}
\frac{\sin\left(\omega - \alpha\_p\right)d}{\left(\omega - \alpha\_p\right)d} - \eta & -i \frac{\cos\left(\omega - \alpha\_p\right)d}{\left(\omega - \alpha\_p\right)d} \\
\frac{\cos\left(\omega - \alpha\_p\right)d}{\left(\omega - \alpha\_p\right)d} & \frac{\sin\left(\omega - \alpha\_p\right)d}{\left(\omega - \alpha\_p\right)d} - \eta
\end{pmatrix} = 0
\tag{36}
$$

The parameters *d*, *η* **(**the Fredholm eigenvalue) and *ωp* (the plasma frequency) can take in principle, arbitrary values but for a specific media can be take numeric values. Now, we remember that we must also impose *η* ¼ 1**.**

Then, Eq. (36) has two resonances:

$$
\alpha\_1 = \frac{\pi}{4d} + \alpha\_p \tag{37}
$$

Ωð Þ þ

Ωð Þ �

C*n*,*e* 2*ωe* j j*p*4*πωe*

*iωet* ∗ P*e Sb* ð Þ*t e*

ð Þ 2ð Þ *ω* � *ωe ωe* is a rectangular function.

*Sa*

*e*,*T Sa*ðÞ¼ *t e*

And

**Figure 1.**

*F* P*e Sa* ð Þ*t e iωet* h i ¼ X∞

*ym*

**9**

Where *p*4*πωe*

**9. Conclusions**

In Eqs. (39) and (40) P*e*

*The two rectangular functions p*4*πωe*

Denoting the Fourier transform like

*On the Zap Integral Operators over Fourier Transforms DOI: http://dx.doi.org/10.5772/intechopen.94573*

*n*¼�∞

And for the convolution we have

*F* P*e Sa* ð Þ*t e*

runners of the Zap projection operators.

Then, the Fourier transform of the GPO is

*e*,*T Sa*ð Þ¼ *t* � *T e*

ð Þ 2ð Þ *ω* � *ωe ωe and p*4*πωu*

*iωet* P*e Sa*

ð Þ 2ð Þ *ω* � *ωu ωu :*

ð Þ *T* � *t* are simple projection operators [11].

*iωut* h i ¼ *FSa* ð Þ *ω* � *ωe FSb* ð Þ *ω* � *ωu* (43)

*F*½ �� *f t*ð Þ Fð Þ *ω* (41)

*iωe*ð Þ *T*�*t* P*e Sa*

ð Þ 2ð Þ *ω* � *ωe ωe e*

Then we see that the set of Fourier transforms of the GPOs behaves like a set of orthogonal basis functions for the frequency domain, that is, the resonant functions

*e* ð Þ **r**;*ω* as we can verify in **Figure 1**. So the GPO can be considered as the fore-

We can conclude that the Zap projection operators (ZPO) can be used as an alternative approach to the generalized projection operators (GPO) that is like an alternative for clean the evanescent signals [10] from disturbances generated by the sources and at the same time to clean the source signals from resonant solutions. We

ð Þ*t* (39)

ð Þ *T* � *t* (40)

�*i*ð Þ *ω*�*ωe πn* � *FSa* ð Þ *ω* � *ωe* (42)

And

$$
\alpha\_1 = \frac{3\pi}{4d} + \alpha\_p \tag{38}
$$

### **8. Forerunners of the zap projection operators**

In Section **6** we defined a new class of integral operators we named Zap projection operators that literally cleans from a broadcasting problem the inhomogeneous part and leaves a projected homogeneous version. These operators act directly over an inhomogeneous Fredholm equation and are related to the Fredholm operators. But recently, we have defined another set of operators we called generalized projection operators (GPO) which projects a complete broadcasting signal (maybe described by a GIFE) not only into several independent mutually orthogonal signals but also can reverse the time direction as we wish. These GPO may be considered as the precursors of the Zap operators and we will see why. We remember their form:

*On the Zap Integral Operators over Fourier Transforms DOI: http://dx.doi.org/10.5772/intechopen.94573*

**Figure 1.** *The two rectangular functions p*4*πωe* ð Þ 2ð Þ *ω* � *ωe ωe and p*4*πωu* ð Þ 2ð Þ *ω* � *ωu ωu :*

$$\mathfrak{Q}\_{e,T}^{(+)}\mathbb{S}\_{a}(t-T) = e^{i\alpha\_{t}t}\mathbb{P}\_{\mathbb{S}\_{a}}^{\epsilon}(t) \tag{39}$$

And

*P* **r**;*ω*; *u*ð Þ∘ ð Þ **r**;*ω* h i

Then, by putting *η* ¼ 1 and Δ ¼ 0 finally

*u*ð Þ¼ **r**;*ω* lim*η*!1

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

*P* **r**; *ω*; *u*ð Þ∘ ð Þ **r**; *ω* h i

taking **K**ð Þ∘ ð Þ *ω* and impose the conditions *η* ¼ 1 and Δ ¼ 0**.**

problem of two steps: homogeneous and inhomogeneous. Now the last condition over the Fredholm determinant is

> sin *ω* � *ωp* � �*d*

> > *ω* � *ωp*

cos *ω* � *ωp* � �*d*

*ω* � *ωp* � �*d*

values. Now, we remember that we must also impose *η* ¼ 1**.**

**8. Forerunners of the zap projection operators**

� �*d* � *η* �*i*

Δ

0

BBB@

Then, Eq. (36) has two resonances:

And

**8**

*i*

½Δð Þ*η η* �X

So it is irrelevant the part of the problem concerning the two sources, it is only a problem about resonances. Our problem is now to find the resonant frequencies by

But, what is the real advantage of the *P* and *Q* operators?, the answer is that the Zap operator formalism may be viewed as a test for distinguish between an expression that cannot be transformed or yes, in whatever sense between the homogeneous and inhomogeneous equations under the rules established above; if not, we can ensure that some kind of irregular things are present. In case of the positive transformation, we have the confidence that both kinds of solutions can coexists, and then we can separate the solutions for convenience as if it was a

2

*αiδ*ð Þ **r** � **r***i* **K**ð Þ∘ ð Þ *ω* **e***i*

!

*u*ð Þ¼ **r**;*ω ye*ð Þ **r**;*ω* (35)

1

4*d* þ *ωp* (37)

4*d* þ *ωp* (38)

CCCA ¼ 0 (36)

þ*η***K**ð Þ∘ ð Þ *ω u*ð Þ� **r**;*ω* (34)

*i*¼1

cos *ω* � *ωp* � �*d*

*ω* � *ωp* � �*d*

sin *ω* � *ωp* � �*d*

*ω* � *ωp* � �*d* � *η*

The parameters *d*, *η* **(**the Fredholm eigenvalue) and *ωp* (the plasma frequency) can take in principle, arbitrary values but for a specific media can be take numeric

*ω*1 ¼ *π*

*ω*1 ¼ 3*π*

In Section **6** we defined a new class of integral operators we named Zap projection operators that literally cleans from a broadcasting problem the inhomogeneous part and leaves a projected homogeneous version. These operators act directly over an inhomogeneous Fredholm equation and are related to the Fredholm operators. But recently, we have defined another set of operators we called generalized projection operators (GPO) which projects a complete broadcasting signal (maybe described by a GIFE) not only into several independent mutually orthogonal signals but also can reverse the time direction as we wish. These GPO may be considered as the precursors of the Zap operators and we will see why. We remember their form:

$$\mathbf{Q}\_{\mathbf{e},T}^{(-)}\mathbf{S}\_{\mathbf{a}}(t) = e^{i\alpha\_{\mathbf{e}}(T-t)}\mathbf{P}\_{\mathbf{S}\_{\mathbf{a}}}^{\mathbf{e}}(T-t) \tag{40}$$

In Eqs. (39) and (40) P*e Sa* ð Þ *T* � *t* are simple projection operators [11]. Denoting the Fourier transform like

$$\bigcirc \mathsf{F}[f(t)] \equiv \mathsf{F}(w) \tag{41}$$

Then, the Fourier transform of the GPO is

$$\mathbb{E}\_{\mathsf{C}}\mathsf{F}\Big[\mathsf{P}\_{\mathsf{S}\_{a}}^{\mathsf{c}}(\mathsf{t})\mathsf{e}^{i\alpha\_{\mathsf{t}}\mathsf{t}}\Big] = \sum\_{n=-\infty}^{\infty} \mathsf{C}\_{n,\mathsf{c}}|\mathsf{2}a\_{\mathsf{c}}|p\_{4\mathsf{z}a\_{\mathsf{c}}}(\mathsf{2}(\mathsf{w}-\mathsf{o}\_{\mathsf{c}})a\_{\mathsf{c}})\mathsf{e}^{-i(\mathsf{o}-\mathsf{o}\_{\mathsf{c}})\mathsf{z}\mathsf{m}} \equiv F\_{\mathsf{S}\_{a}}(\mathsf{w}-\mathsf{o}\_{\mathsf{c}}) \tag{42}$$

Where *p*4*πωe* ð Þ 2ð Þ *ω* � *ωe ωe* is a rectangular function. And for the convolution we have

$$\mathbb{E}\_{\mathbb{S}}\mathfrak{p}^{\varepsilon}\Big[\mathbf{P}\_{\mathcal{S}\_{a}}^{\varepsilon}(t)\mathfrak{e}^{ia\_{\mathbb{H}}t}\*\mathbf{P}\_{\mathcal{S}\_{b}}^{\varepsilon}(t)\mathfrak{e}^{ia\_{\mathbb{H}}t}\Big] = F\_{\mathcal{S}\_{a}}(\boldsymbol{\alpha}-\boldsymbol{\alpha}\_{\varepsilon})F\_{\mathcal{S}\_{b}}(\boldsymbol{\alpha}-\boldsymbol{\alpha}\_{\mathbb{H}})\tag{43}$$

Then we see that the set of Fourier transforms of the GPOs behaves like a set of orthogonal basis functions for the frequency domain, that is, the resonant functions *ym e* ð Þ **r**;*ω* as we can verify in **Figure 1**. So the GPO can be considered as the forerunners of the Zap projection operators.

## **9. Conclusions**

We can conclude that the Zap projection operators (ZPO) can be used as an alternative approach to the generalized projection operators (GPO) that is like an alternative for clean the evanescent signals [10] from disturbances generated by the sources and at the same time to clean the source signals from resonant solutions. We can also use the two classes of projectors in a consecutively manner. The former vision suppose that the evanescent waves [10] can be considered as part of the conventional traveling waves like an everything and that we must take away the effect of the resonances with the application of the *Q* operator. In any case we have shown the power of the Fourier transform applied to mathematical analysis in broadcasting problems and to physically characterize and solve them.

**References**

[1] J. M. Velázquez-Arcos, *Fredholm's equations for subwavelength focusing*. J. Math. Phys. Vol. 53, No. 10, 103520 (2012), doi: 10.1063/1.4759502.

*On the Zap Integral Operators over Fourier Transforms DOI: http://dx.doi.org/10.5772/intechopen.94573*

International Conference pp.411–414,

[10] Xiang-kun Kong, Shao-bin Liu, Haifeng Zhang, Bo-rui Bian, Hai-ming Li et al., *Evanescent wave decomposition in a novel resonator comprising unmagnetized and magnetized plasma layers*, Physics of Plasmas, Vol. 20, 043515 (2013); doi:

[11] J. M. Velázquez-Arcos, J. Granados-Samaniego, A. Cid-Reborido and C. A. Vargas,*The electromagnetic Resonant Vector and the Generalized Projection Operator,* IEEE Xplore, 2018 Progress In Electromagnetics Research Symposium (PIERS-Toyama), Japan, pp 1225–1232.

[12] F. Hernández-Bautista, C. A. Vargas and J. M. Velázquez-Arcos, *Negative refractive index in split ring resonators*, Rev. Mex. Fis. Vol. **59**, no. 1, pp. 139– 144, January–February 2013, ISSN:

[13] von der Heydt von N. *Die*

[14] Richard R. Goldberg, *Fourier Transforms, Cambridge University Press*. ISBN-13: 978–0521095556; ISBN-10:

[15] David Brandwood, *Fourier Transforms in Radar and Signal*

ISBN-10:<PHONE_NUMBER>; ISBN-13:

*Processing*, Artech House Radar Library

*Schrödinger-Gleichung mit nichtlokalem Potential. I.) Die Resolvente*, Annalen der Physik *Volume 29, Issue 4, 1973, PP 309–*

9–13 Sept. 2013 doi: 10.1109/ ICEAA.2013.6632268.

10.1063/1.4802807.

0035-00IX.

0521095557.

978–1580531740.

*324.*

[2] Harry Hochstadt, *Integral Equations*, Wiley Classics Library, ISBN-10 :<PHONE_NUMBER>; ISBN-13 : 978–1580531740 .

[3] Witold Pogorzelsky and Ian Snnedon, *Integral Equations and their Applications*, Pergamon, ISBN-10 :<PHONE_NUMBER>; ISBN-

[4] A. Mondragón, E. Hernández and J. M. Velázquez Arcos *Resonances and Gamow States in Non-Local Potentials*, Annalen der Physik *Volume 48, Issue 8,*

[5] R. de la Madrid,*The decay widths, the decay constants, and the branching fractions of a resonant state*, Nuclear Physics A 940 (2015) 297–310.

[6] J. M. Velázquez-Arcos, C. A. Vargas, J. L. Fernández-Chapou, A. L. Salas-Brito, *On computing the trace of the kernel of the homogeneous Fredholm's equation*. J. Math. Phys. Vol. 49, 103508 (2008), doi:

[7] R. de la Madrid,*The rigged Hilbert space approach to the Gamow states*, J. Math. Phys. Vol. **53**, No. 10, 102113 (2012), doi: 10.1063/1.4758925.

[8] J. M. Velázquez-Arcos and J.

Electronics and Communication Engineering (IOSR-JECE)*e-ISSN: 2278– 2834, p- ISSN: 2278–8735. Volume 11, Issue 2, Ver. I (Mar-Apr. 2016), PP*

*42-48www.iosrjournals.org.*

**11**

Granados-Samaniego, *Wave propagation under confinement break*, IOSR Journal of

[9] J. M. Velázquez-Arcos, J. Granados-Samaniego, C.A. Vargas,*The confinement of electromagnetic waves and Fredholm's alternative*, Electromagnetics in

Advanced Applications (ICEAA), 2013

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1991, PP 503–616 .

10.1063/1.3003062.


**Abstract**

Disparities in older adults' care and experiences in long-term care facilities (LTCFs) such as nursing homes and assisted living/residential care communities reflect disparities in the broader society. Various policies and institutional practices related to economic opportunity, education, housing, health care, and retirement financing have created and maintain inequitable social structures in the United States. This chapter describes racial and ethnic disparities among older adults in LTCFs in the United States and the systemic factors associated with those disparities. It presents a conceptual framework for understanding the role of structural racism in the racial and ethnic inequities experienced by LTCF residents. In the framework, structural racism directly contributes to racial and ethnic inequities among LTCF residents through LTCF-related policies and practices. Structural racism also indirectly causes disparities among LTCF residents through health and economic disparities. The chapter describes current efforts that address the effects of structural racism within LTCFs and concludes with practice and policy recommendations to redress racial and ethnic disparities among LTCF residents.

**Keywords:** long-term care facilities, nursing homes, assisted living communities, health disparities, racial and ethnic inequities, Medicaid policy, Long-Term Care Ombudsman Program

## **1. Introduction**

Structural racism affects individuals and communities across the life course. For older Americans, inequities in health access, quality, and outcomes caused by racism and systemic barriers in the United States can be exacerbated in later life in a variety of domains including physical and cognitive health, mortality rates, and quality of care. Systems for care in later life include long-term care facilities (LTCFs) such as nursing homes and assisted living/residential care communities. Paired with the demographic trend of increasing proportions of older adults from historically minority racial and ethnic groups [1] is a growing utilization of LTCFs by people of color [2]. Unfortunately, older adults of color in the United States experience disparities in access to quality nursing homes; access to care in assisted living communities; quality of care and quality of life in LTCFs; health outcomes as LTCF residents; and social engagement within LTCFs. These disparities are associated with a variety of structural factors (e.g., federal and state policy related to LTCF funding and oversight, housing policies that have created racially segregated communities, and workforce practices that lead to income and wealth disparities). The growing number of people of color in LTCFs and persistent disparities within them creates an urgency to address racial and ethnic inequities in quality of care and quality of life for older adults of color living in LTCFs.

#### **1.1 Long-term care facilities (LTCFs)**

Older adults who experience chronic limitations in physical and cognitive functioning may need long-term services and supports. Long-term care encompasses a range of services and supports that assists individuals in completing activities such as dressing, preparing meals, medication management, and housework [3]. Most long-term care is provided at home by family caregivers [4]; however, long-term care is also available in long-term care facilities (LTCFs). The need for long-term services and supports increases as individuals age, as does the likelihood of not having the assistance of a spouse who can provide informal care. For this reason, and due to the aging of the population in the United States, a growing number of older adults are utilizing LTCFs [3, 5].

In the United States, the majority of the funding for long-term services and supports comes from public sources, but many people privately pay or use private long-term care insurance [6, 7]. Medicaid, a means-tested program, is the primary funder of care in LTCFs. The federal and state governments jointly fund Medicaid, but it is administered by the states. Each state sets its own eligibility requirements for Medicaid, which include income and resource limits. In contrast, Medicare is administered at the federal level, and eligibility requirements are tied to eligibility for Social Security or Railroad Retirement benefits [8].

#### *1.1.1 Nursing homes*

Nursing homes are residential communities that provide a higher level of care than can often be provided at home or through other community-based services. Nursing homes may also provide health care services such as physical or occupational therapy to help patients recover from illnesses or injuries. The median monthly U.S. nursing home cost in 2020 was \$8,821 for a private room and \$7,756 for a semi-private room [9]. Most nursing home residents pay for long-term nursing home care with Medicaid, with Medicare paying for more short-term post-acute nursing care in skilled nursing facilities [10].

Private nursing homes became common in the United States beginning in the late 1930s, after the Social Security Act of 1935 prohibited older adults who lived in public alms houses from receiving Old Age Assistance [11]. Wealthier White older adults were able to afford private nursing home care; however, this option was financially inaccessible for poorer White people and poorer people of color [12]. Public funding for nursing home care was not available until the 1950s [13]. These policy decisions created financial barriers for people of color, particularly African Americans, to access nursing home care.

In contrast to the past, today older adults of color are overrepresented in the nursing home population, representing approximately 25% of nursing home residents [2, 10]. The trend for increasing portions of residents of color in nursing homes seems to be driven in part by White older adults disproportionately accessing more appealing alternatives to nursing homes that are funded by Medicaid waivers for Home and

#### *Addressing Systemic Factors Related to Racial and Ethnic Disparities among Older Adults… DOI: http://dx.doi.org/10.5772/intechopen.99926*

Community-Based Services [2] and privately paying for care in assisted living communities [14]. At the same time that an increasing percentage of people of color are using nursing homes, there have been increased closures of nursing homes across the country, with closures concentrated in disadvantaged communities of color [2].

Nursing homes tend to be quite segregated by race and ethnicity [15], a phenomenon related to past structural racism. Policies such as the 1946 Hill-Burton Act (which funded construction of "separate but equal" nursing homes) and southern Jim Crow laws combined with discriminatory practices in hospital discharge planning and nursing home admissions to create and maintain segregated nursing home systems [12, 16, 17]. In the 1960s, the Johnson administration failed to use provisions of the Civil Rights Act to desegregate nursing homes and prohibit discrimination in nursing home practices [12, 17, 18]. Housing policies such as redlining created and perpetuated racial segregation of neighborhoods which in turn supported racial segregation of nursing homes, as nursing home residents tend to come from their surrounding communities [15].

#### *1.1.2 Assisted living/residential care communities*

Assisted living or similar residential care communities are another type of LTCF. They serve older adults who cannot live alone safely, but do not need the level of care provided at nursing homes. They offer personal care and household assistance to residents in a homelike environment. Assisted living and residential care communities can range from small homes with a few residents to large communities of private apartments in large residential settings, which tend to be chain-affiliated and owned by for-profit companies. These communities generally provide communal meals and opportunities for socialization and physical activities in addition to personal care services. Assisted living communities tend to be in urban/suburban areas and communities characterized by high levels of education, income, and financial resources [2]. Licensing of assisted living/residential care communities is at the state level, with variations across the states.

Many Americans have a more favorable impression of assisted living than of nursing homes, and it the fastest growing model of residential long-term care [19]. The 2020 median monthly cost of assisted living care was \$4,300 – substantially less than care in a nursing home [9] – but prohibitive for many to pay out of pocket. Medicaid only covers assisted living in states that have Medicaid waivers for Home and Community-Based Services that fund assisted living [20]. Although most states have these waivers, the coverage is low, and smaller and poorer states are less likely to adopt Medicaid waivers [21]. Furthermore, Medicaid eligibility, benefits, cost sharing requirements, and reimbursement rates vary by state [22], and evidence suggests that racial bias within a state is related to lower levels of Medicaid spending [23]. A few states do not provide any Medicaid funding for assisted living/residential care and in others, Medicaid covers personal care, but not room and board. In states that do fund assisted living with Medicaid, low reimbursement rates and the costs of administering Medicaid deter many assisted living providers from becoming Medicaid certified [24]. Indeed, less than half of the assisted living/residential care communities in the United States accept Medicaid [10]. As might be suggested by these systemic barriers, older adults of color are underrepresented in assisted living communities [10, 25].

### **2. Racial and ethnic disparities in U.S. long-term care facilities**

The almost half-million older adults of color who currently live in U.S. LTCFs [3] face disparities along a variety of dimensions including health outcomes, quality

of care, quality of life, and social integration compared to non-Hispanic White residents. Much of the evidence of racial and ethnic disparities in long-term care comes from nursing homes, which are federally mandated to provide detailed health outcome and demographic data for their residents. This evidence points to racial and ethnic disparities in health and quality of life outcomes, engagement with health services, and access to quality care.

In nursing homes, health outcome disparities are evidenced by findings that Black residents have a higher risk for developing pressure ulcers [26, 27] which can lead to serious medical complications, and are less likely to recover from pressure ulcers present when they are admitted [28]. There are ample examples of racial and ethnic disparities in engagement with health services and health care quality within LTCFs. Black residents have received less pain management [29], have been subject to more use of physical restraints [30], and are less likely to receive a flu vaccine [31] compared to White residents. Black residents and those categorized on medical records as coming from "other" racial groups (e.g., American Indian/Alaska Native; Native Hawaiian/Pacific Islander) were found to be less likely to have toileting plans for incontinence than White residents [29]. Depressive symptoms – which can have severe mental health consequences if depression is left untreated – seem to be underreported for Black, Latinx, and Asian nursing home residents [32].

Racial and ethnic disparities in quality of life outcomes such as cultural fit and social engagement have also been reported. For example, higher proportions of minority residents in nursing homes are associated with more quality of life deficiencies reported in the facility [33]. Chinese residents have reported a lack of culturally appropriate food, which related not only to their feelings of belonging and being valued, but also to receiving enough nutrition [34]. Compared to White residents, Black, Latinx, and other nursing home residents of color have scored lower on social engagement measures that include interacting with others, accepting invitations to group activities, being at ease in group/structured activities, and establishing their own social goals [35]. Indeed, nursing home residents of color have reported lower quality of life indicators than White residents across multiple domains, including personal attention, food, engagement within the facility and with staff, and mood [36].

Data regarding complaints received by the U.S. Long-Term Care Ombudsman Program extends our understanding of racial and ethnic disparities in LTCFs to include assisted living communities. The Long-Term Care Ombudsman Program is a federally mandated program administered at the state level that advocates for LTCF residents in both nursing homes and assisted living communities. Local ombudsmen conduct site visits, make referrals as needed, provide resident and public education, engage in policy advocacy, and receive and resolve complaints on behalf of residents. In their role as resident advocate, state Ombudsman Programs are well positioned to enhance our understanding of racial and ethnic disparities among LTCF residents. However, State Ombudsman Programs are not required to collect and report data about the race and ethnicity of the residents for whom they receive complaints; they are only required to report aggregate-level race and ethnicity data for the facilities under their purview.

A recent study of ombudsman complaints in the Dallas, TX, area collected race/ ethnicity data associated with resident complaints in an examination of racial and ethnic differences in complaint types and resolution rates [37]. Residents of color were more likely than White residents to file complaints related to residents' rights (i.e., abuse, access to information, autonomy, financial rights). Interestingly, complaints more likely to be resolved in nursing homes and assisted living communities with higher percentages of minority residents; however, this finding was related to the resolution of complaints from or on behalf of White residents living

*Addressing Systemic Factors Related to Racial and Ethnic Disparities among Older Adults… DOI: http://dx.doi.org/10.5772/intechopen.99926*

in those communities [37]. In focus groups, ombudsmen noted they had witnessed residents of color who refrained from making complaints about care compared to complaints about rights for fears of retaliation or being branded as a problem in the community. The ombudsmen also described ways in which LTCFs did not provide culturally appropriate environments for all residents (e.g., staff who could not communicate with residents in their language). Finally, the ombudsmen provided additional information about staffing ratios at Medicaid-certified facilities noting at times that only one aid would be available to care for a dozen residents needing aid.

### **2.1 Between- and within-facility sources of racial and ethnic disparities in LTCFs**

As described earlier in this chapter, LTCFs tended to be racially segregated which relates to disparities in access to quality LTCF care. Many of the racial and ethnic disparities LTCF residents experience arise from differences between LTCFs that serve higher percentages of residents of color, particularly Black residents, and those that serve lower percentages [27, 33, 36]. LTCFs that serve higher percentages of residents of color tend to have fewer financial and community resources and insufficient staffing, with a correspondingly high number of care deficiencies, inadequate direct care, and low quality of care ratings [33, 38–40]. Economic factors play a major role in these differences. In general, LTCFs with higher concentrations of residents of color rely more on Medicaid funding than LTCFs serving predominantly White residents and are therefore more constrained by Medicaid's lower reimbursement rates [33, 38, 39]. Indeed, the more Medicaid-reliant a nursing home is, the fewer resources it has to devote to resident-directed care and activities, improving the home environment, and other quality of life and quality of care related pursuits [41].

Although facility-level differences account for many of the racial and ethnic disparities among nursing home residents, disparities still exist within individual facilities such as in vaccination rates and quality of care [31, 42]. This can be attributed in part to an unconscious provider bias, which can lead to health care providers limiting the amount of information they share with residents of color and result in less patient-centered communication [43]. It can also be related to the fact that people of color tend to be admitted to nursing homes with worse health and greater care needs [44].

#### **2.2 COVID-19 and racial and ethnic disparities in long-term care facilities**

The COVID-19 pandemic ushered in a heightened awareness of structural racism and discrimination related to the provision of health care to older adults. Communities of color were disproportionately affected by COVID-19 infections, severe illness, and deaths [45]. The Centers for Disease Control and Prevention [46] reported that approximately 22% of the COVID-19 deaths in the United States in 2020 occurred in LTCFs. Prior to the pandemic about 63 percent of nursing homes had infection-control deficiencies [47]. Because older people of color were overrepresented in nursing home populations in general – and specifically more likely to reside in lower-quality nursing homes – this put them at an increased risk for contracting infectious diseases like COVID-19. Indeed, facility-level disparities quickly became apparent. In the early months of the pandemic in the United States, *The New York Times* [48] reported that nursing homes with higher percentages of Black or Latinx residents were twice as likely to report COVID-19 infections than those with predominantly White residents. A subsequent analysis of Centers for Medicare & Medicaid Services data through May 2020 had similar findings [49].

Nursing homes with higher portions of residents of color tended to be in areas with higher levels of COVID-19 cases and deaths. It also found that LTCFs with higher proportions of residents of color were more likely than those with low proportions of residents of color to experience COVID-19 infections and deaths and report a shortage of aids during the pandemic [49].


*4.2. Western Blot Analysis*

Cultured cells were lysed in lysis buffer (50 mM Tris pH 7.4, 150 mM NaCl, 2 mM EDTA, 0.8% NP-40, 0.2% Triton X-100, 3% glycerol). Cell lysates were subjected to polyacrylamide gel electrophoresis (PAGE), and proteins were transferred onto a polyvinylidene fluoride (PVDF) membrane. Membranes were soaked in 5% nonfat milk or 5% BSA solution for one hour to block nonspecific binding of proteins, and then incubated with primary antibodies overnight at 4 ◦C. On the following day, membranes were incubated with secondary antibodies for 2 h at room temperature, and WesternBright ECL (Advansta, Menlo Park, CA) was used with the luminescent image analyzer (Jun Yi Dong Fang, Beijing, China) to capture images. Uncropped scans can be found in Figure S1.

Antibodies used in western blotting were: KLF5 (1:1000, 21017-1-AP, Proteintech), AR (1:1000, 5153S, Cell Signaling), PSA (1:3000, 10679-1-AP, Proteintech), MYC (1:1000, 9402, Cell Signaling), cyclin D1 (1:10000, ab134175, Abcam), FLAG (1:3000, SAB4200071, Sigma), HA (1:3000, 3724S, Cell Signaling), GAPDH (1:3000, 60004-1-Ig, Proteintech).


**4. Simulation Results**

The EO is applied to determine the optimal operation of MG to achieve technical and economic benefits with respecting the associated operational and emission costs for two distribution systems. For the optimal operation, the control variables are the active power resulted from different RERs, including BGs, wind, and PV, and the power factor of BGs.


Chapter 8 **Fracture Toughness Determination with the Use of Miniaturized Specimens 143** Jan Dzugan, Pavel Konopik and Martin Rund

Preface

During recent years, the contact mechanics and fracture mechanics have found a considera‐ ble application in the solution of engineering problems to increase the life of the component. Contact mechanics studies the stress and strain states of bodies in contact; it is a contact that leads to friction interaction and wear. In recent years, computational contact mechanics has been a topic of intense research. The aim of this research is to devise robust solution schemes and new discretization techniques for the description of contact phenomena, which can then be applied to a much broader range of engineering analysis areas than is currently the case. The focus will be on a detailed treatment of the theoretical formulation of contact problems with regard to mechanics and mathematics. Fracture is understood to be the sepa‐ ration of a body of material into two or more pieces, whereby the load carrying is reduced to zero. The process of fracture can be considered to be made up of two components, crack initiation and crack propagation. Fracture mechanics has developed into a useful tool in the design of crack-tolerant structures and in fracture control; it also has a place in failure analy‐ sis. Fracture mechanics makes it possible to determine whether a crack of a given length in a material of known fracture toughness is dangerous because it will propagate to fracture at a given stress level. If the cause of crack extension may not be controlled, the only thing left to

The different contributions of this book will cover the various advanced topics of research. It provides some needed background with respect to contact mechanics, fracture mechanics, and the use of finite element methods in both. All the covered chapters of this book are of a theoreti‐ cal and applied nature, suitable for the researchers of engineering, physics, applied mathemat‐ ics and mechanics with an interest in computer simulation of contact and fracture problems. This book contains two sections as its name; Chapters 1–7 deal with contact mechanics, and Chapters 8–13 deal with fracture mechanics. Hermetic sealing studies are carried out in Chapter 1. Sealing capacity depends on the contact characteristics—the relative contact area and the gap density in the joint. In this chapter, the contact of a single asperity is considered taking into account the influence of the remaining contacting asperities. The response of a nanometer-scaled single asperity onto flat surfaces is experimentally accessible using atomic force microscopy, which is studied in Chapter 2. The author describes three experimental methods based on atomic force microscopy and corresponding methods for statistical data analysis. Chapter 3 presents an update of theories involving the differential hardness prob‐ lem, starting from the hypothesis made by Tabor for the contact between a sphere and a

designer is to calculate the critical length in advance.


**1. Introduction**

Plasma exchange (PEX) is an invasive therapeutic method, separating plasma from blood cells. Thus, pathogenic antibodies or other large molecules are removed and plasma is replaced by human albumin and/or fresh frozen plasma (FFP). The method was first developed in the first half of the twentieth century. Over the years a significant improvement in the PEX technique, patient safety and broadening of indications were observed. Selective techniques were also introduced into practice, leading to selective removal of proteins and reduction of protein loss during the standard procedure, especially fibrinogen. Thus, improved effectiveness and patient safety was achieved.


**Author details**

Annalisa Fogazzi, Fabianna Timelli, Annalisa Vezzoli, Valentina Tradati and Rita Padoan\* Regional Support Cystic Fibrosis Centre, University of Brescia—Children's Hospital, Brescia, Italy

\*Address all correspondence to<EMAIL_ADDRESS>
© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Chapter 7**

**Abstract**

self-management

**1. Introduction**

diet, and exercise.

**105**

Designing Heterogeneous-mHealth

In this chapter, we will discuss the design and development of a patient passport mHealth application for Cystic Fibrosis adults from ideation to app-store release. By allowing the patients access to their own unique data, it is anticipated that it will be of benefit when travelling abroad and between CF centres. The design process followed a pipeline we developed that is informed by patient and healthcare professional input. The app structure resembles an Irish patient file and is divided into three categories: "My CF Info", "My Medical History", and "My Clinical Appointments". My CF Info allows the patient to store personal information such genotype, medical team contact information, physiotherapy, allergies, and medications. My Medical History allows the user to record information such as CF renal disease, CF diabetes, and the insertion/removal of a portacath/gastrostomy tube. My Clinical Appointments allows the user to record the type of appointment (annual assessment, clinic, other) and all information that would ordinarily be inserted into a patient file such as weight, height, spirometry and other comments. Weight and lung function are also displayed in a plot graph. The app has undergone pilot testing

Apps for Cystic Fibrosis Adults

*Mairead McCarthy, Sabin Tabirca and Barry J. Plant*

with five CF adults before being rolled out onto the Google Play Store.

Cystic Fibrosis (CF) is the most common life limiting genetic disease affecting Caucasians. Patients must adhere to rigorous therapies in order to manage their condition. Such therapies include airway clearance physiotherapy, medications,

Ireland has the highest prevalence of CF worldwide, with 1 in 19 people being carriers for the CF gene [1]. Despite this, the life expectancy of patients with CF is rising. A child born with CF in 2009 is now predicted to live into their 50s [2, 3], compared to children born in 1950, of which half would live to the age of 16 and only 3% to 25 years old [4, 5]. However, this increase in life expectancy comes with an increase in patient population. There are now more adult CF patients than pediatric patients in Ireland [6]. It is projected that the pediatric service will need to

grow by 25% and the adult service by 75% by 2025 [7]. This growth in adult population and the advent of ICT technologies in the medical space has led to mobile health applications (mHealth apps) being identified as potential powerful

**Keywords:** cystic fibrosis, patient passport, mHealth, digital health,

*Tamara Vagg, Cathy Shortt, Claire Fleming,*


**3.4 Lipids**

274 Thermoplastic Elastomers

solution due to the temperature and specific mechanical energy to which the product is

One of the main applications of extrusion in high protein content foods is protein texturization. Texturization processes by extrusion can be used to obtain products that imitate the texture, taste, and appearance of meat or seafood with high nutritional value

The use of raw materials with high protein contents in extrusion began around the 1970s, with the use of soy for the production of texturized soy products and meat analogues

In extrusion, the proteins that have been found to form a continuous structure are globular proteins from oilseeds such as soybeans, sunflower seeds, common beans, peas and cottonseed and from cereals, especially wheat gluten proteins (Riaz, 2000; Strahm, 2006).

The extrusion process, physically, converts protein bodies into a homogeneous matrix, while chemically, the process recombines storage proteins in some way into structured fibres (Stanley, 1998). Low moisture (up to 35%) extrusion of vegetable protein can be used to elaborate products to partially or totally substitute meat. Usually, these products are expanded and need to be re-hydrated before consumption. On the other hand, high moisture (>50%) extrusion results in products that do not need to be re-hydrated and can be consumed directly. In general, dry extrusion is applied when the aim is to produce meat extenders and wet extrusion is used for meat analogues (Noguchi, 1998). In dry extrusion, when the conditioned material passes through the die at a high temperature, the water in the material is changed into superheated steam, which expands the extrudate immediately. Water also makes the extrudate very soft, by reducing its viscosity drastically, so the material just after the die is not self-supporting. Therefore, cooling at the die is essential to increase the viscosity of the hot melt and reduce its fluidity so that the necessary pressure and temperature before the die can be maintained. When cooling is done appropriately, the correct amount of extrudate elasticity and fluidity can be obtained to allow a continuous "rope" structure without explosive puffing and the destruction of product integrity (Noguchi, 1998). The mechanism for structure creation with proteins is similar to that with starch in the sense that proteins must be dispersed from their native bodies into a free flowing continuous mass. Texturization occurs between the molecules as they flow in the streamlines to form laminar cross-linked products. Evaporation of water in the mass creates gas bubbles that form alveolar structures held in place by cross-linking in the protein layers

Denaturation during the extrusion process of proteins results in reduction of protein solubility, favours digestibility and inactivates antinutritional factors (such as antitrypsin factor, lectins, etc.). Also, the extrusion of soy protein reduces the bitter taste and the undesirable volatile compounds related to this protein (Areas, 1992; Kitabatake & Doi, 1992). During extrusion, protein structures are disrupted and altered under high shear, pressure, and temperature (Harper, 1984). In the extrusion of proteins, disulfide bonds are cleaved and undergo reorganization and polymerization. Disulfide bonds, non-specific hydrophobic and electrostatic interactions are the main bonds and interactions responsible for protein texturization by extrusion (Areas, 1992). Protein solubility decreases and cross-linking

submitted (Camire, 2000).

(Ledward & Mitchell, 1988; Mitchell & Areas, 1992).

(Cheftel et al., 1992).

(Guy, 2001).

Fats and oils can be described as lipids. Lipids have a powerful influence in extrusion cooking processes by acting as lubricants, because they reduce the friction between particles in the mix and between the screw and barrel surfaces and the fluid melt (Guy, 2001). In the extruder, fats and oils become liquid at temperatures > 40°C, being mixed with the other materials, and are rapidly dispersed as fine oil droplets.

The presence of lipids in quantities lower than 3% does not affect expansion properties, however, in amounts above 5%, reduction in expansion rate is considerable (Harper, 1994). Collona et al. (1998) suggest that the increase in lipid content can be corrected through the reduction in conditioning moisture content, so as not to affect the expansion index of second generation products (directly expanded snacks).

The type of starch and lipid present in the raw material influences the formation of the amylose-lipid complex, with free fatty acids and monoglycerides being more favourable to the formation of this complex than triglycerides (Mitchel & Areas, 1992; Harper, 1994; Camire, 2000).

Moreover, in wet protein extrusion, the presence of lipids does not support protein fibre formation since the lubricating effect of lipids decreases the shear effects and particle alignment (Akdogan, 1999).

#### **3.5 Fibres**

The term "fibres" covers a great variety of substances with different physical, chemical and physiological properties. Dietary fibre consists of fractions of vegetable cells, polysaccharides, lignin, and associated substances, which are resistant to hydrolysis by enzymes present in the digestive system of humans; however, some types of fibres may be

Thermoplastic Extrusion in Food Processing 277

increase in extrusion temperature; however, the effect of moisture is more significant

Water acts as a plasticizer for the starchy material that displaces itself within the extruder, reducing viscosity and mechanical energy, producing higher density products and inhibiting bubble growth. Studies carried out with corn grits demonstrated that expansion is inversely proportional to the moisture content of the material being extruded (Chinnaswamy, 1993; Colonna et al., 1998). With higher moisture, starch gelatinization is reduced and bubble growth is retarded, resulting in denser and less crunchy final products

Inside the extruder, the product that contains molten starch in its composition, when leaving the extruder, has part of its water rapidly evaporated. This water loss is of 3 to 5% and contributes to cooling the product. Subsequent cooling occurs more slowly due to the low thermal conductivity of the extrudate. Also, when emerging from the die, the extrudate undergoes an abrupt pressure fall that also contributes to its expansion (Colonna et al., 1998). The expanded final product presents air cells that are formed due to superheated water vapour pressure. As the temperature of the extrudate is reduced below its glass transition temperature (Tg), it solidifies and maintains its expanded form (Riaz, 2000).

In high moisture extrudates, expansion occurs when the product exits the die, but the structure collapses before the necessary cooling, resulting in a dense and hard product

Another important parameter for extrudate expansion is process temperature. Products do not expand if the temperature does not reach 100°C. Expansion increases with the increase in temperature when moisture content of the material is close to 20%, due to lower viscosity, permitting a more rapid expansion of the molten mass, or due to an increase in water vapour pressure. At low extrusion temperatures, expansion is reduced because starch is not completely molten. Radial expansion degree is proportional to temperature up to a certain value, decreasing at much higher temperatures. The reduction of expansion at very high temperatures is attributed to an increase in dextrinization, weakening starch structure

At high temperatures, the gel is more elastic, forming a matrix with small uniform cells, while at low moisture, the gel formed is not very elastic and the extruded material has large, not very uniform cells. It is expected that an increase in temperature should reduce viscosity of the molten material, favouring bubble growth and producing low density extrudates,

In high moisture extrusion, the properties of the protein extrudates are strongly influenced by extruder conditions (Thiebaud et al., 1996; Noguchi, 1998). Hayashi et al. (1991) reported that extruder barrel temperature was the most important parameter for the texturization of dehulled soybean. Melt temperature is a critical factor in protein cross-linking reactions. Increasing temperature from 140 to 180°C results in a proportional decrease in disulfide linkages formed in extruded soy protein isolates (Areas, 1992). Temperatures lower than 90°C hinder expansion and layer formation (Cheftel et al., 1992). At a given temperature, higher moisture contents result in softer and less texturized extrudates due to reduced

with finer cells and greater crunchiness (Ding et al., 2005).

(Harper, 1994).

(Ding et al., 2005).

(Harper, 1994).

(Colonna et al., 1998).

fermented by bacteria in the colon. As many physiological effects of fibres seem to be related to their solubility in water, they are frequently classified as "soluble" and "insoluble" (Roberfroid, 1993; Stark & Madar, 1994).

Soluble fibres form a gel network or a viscous network, in determined physicochemical conditions, and thus bond water increasing viscosity, retarding gastric transit, reducing glucose, lipid and sterol absorption rates (Gorinstein et al., 2001). Soluble fibres are also seen as fermentable substrates, as they can modify the pH and the microflora of the colon, leading to a reduction or modification of mutagenic agents (Thebaudin et al., 1997). Insoluble fibres increase faecal volume, thus diluting its contents, which reduces the interaction between the intestinal mucosa and any carcinogenic component present. Apart from this, insoluble fibres reduce intestinal transit time, avoiding that mutagenic agents in the faeces interact with the intestinal epithelium (Thebaudin et al., 1997).

Research has shown that cooking fibres by extrusion can produce changes in their structural characteristics and physicochemical properties, with the main effect being a redistribution of insoluble fibre to soluble fibre (Camire et al., 1990; Guillon et al., 1992; Larrea et al., 2005). This effect would be the result of the rupture of covalent and non-covalent bonds between carbohydrates and proteins associated to the fibre, resulting in smaller molecular fragments, that would be more soluble (Fornal et al., 1987; Wang et al., 1993).

Various researchers have reported a reduction in expansion index (EI) when dietary fibre is added to the formulation (Hsieh et al., 1989; Ilo et al., 1999; Vernaza et al., 2009). The reduction in the expansion index due to fibre addition can be explained through different mechanisms: (i) fibrous materials found in the formulation of extruded products include materials composed of hemicellulose, cellulose and lignin. In normal extrusion conditions, these materials tend to remain firm and stable during processing, without size reduction. The physical presence of fibres in air cell walls reduces the expansion potential of the starchy film (Guy, 2001); larger particles, such as bran, tend to rupture air cell walls of the extruded product, causing a reduction in expansion index (Riaz, 2000); (ii) according to Colonna et al. (1998), maximum degree of expansion is closely related to starch content, with maximum expansion being obtained for pure starches. As bran contains high fibre content, it reduces the starch content of the formulations; (iii) non-starch polysaccharides, such as fibres, may bind water more strongly than proteins and starch during extrusion. This water binding capacity inhibits water loss at the die, that is, at the exit of the extruder, reducing expansion (Camire & King, 1991); (iv) the starch present cannot be totally gelatinized in the presence of fibre and is thus not capable of supporting expansion (Camire & King, 1991); and (v) the porous structure of the extrudate depends of the plasticity of the mass before the die, for which starch is mainly responsible. Porosity, defined by the existence of fine pores and a tender structure, is influenced by alterations in the plasticity of the mass, affected by the composition of the mix. Formulations can be enriched by plasticizing substances or by non-plasticizing substances that retard expansion by diluting starch, as is the case of fibres (Colonna et al., 1998).

#### **3.6 Moisture and temperature**

In the extrusion process of expanded products with low moisture, the expansion of the final product is inversely related to the moisture of the raw material and directly related to the

fermented by bacteria in the colon. As many physiological effects of fibres seem to be related to their solubility in water, they are frequently classified as "soluble" and "insoluble"

Soluble fibres form a gel network or a viscous network, in determined physicochemical conditions, and thus bond water increasing viscosity, retarding gastric transit, reducing glucose, lipid and sterol absorption rates (Gorinstein et al., 2001). Soluble fibres are also seen as fermentable substrates, as they can modify the pH and the microflora of the colon, leading to a reduction or modification of mutagenic agents (Thebaudin et al., 1997). Insoluble fibres increase faecal volume, thus diluting its contents, which reduces the interaction between the intestinal mucosa and any carcinogenic component present. Apart from this, insoluble fibres reduce intestinal transit time, avoiding that mutagenic agents in

Research has shown that cooking fibres by extrusion can produce changes in their structural characteristics and physicochemical properties, with the main effect being a redistribution of insoluble fibre to soluble fibre (Camire et al., 1990; Guillon et al., 1992; Larrea et al., 2005). This effect would be the result of the rupture of covalent and non-covalent bonds between carbohydrates and proteins associated to the fibre, resulting in smaller molecular fragments,

Various researchers have reported a reduction in expansion index (EI) when dietary fibre is added to the formulation (Hsieh et al., 1989; Ilo et al., 1999; Vernaza et al., 2009). The reduction in the expansion index due to fibre addition can be explained through different mechanisms: (i) fibrous materials found in the formulation of extruded products include materials composed of hemicellulose, cellulose and lignin. In normal extrusion conditions, these materials tend to remain firm and stable during processing, without size reduction. The physical presence of fibres in air cell walls reduces the expansion potential of the starchy film (Guy, 2001); larger particles, such as bran, tend to rupture air cell walls of the extruded product, causing a reduction in expansion index (Riaz, 2000); (ii) according to Colonna et al. (1998), maximum degree of expansion is closely related to starch content, with maximum expansion being obtained for pure starches. As bran contains high fibre content, it reduces the starch content of the formulations; (iii) non-starch polysaccharides, such as fibres, may bind water more strongly than proteins and starch during extrusion. This water binding capacity inhibits water loss at the die, that is, at the exit of the extruder, reducing expansion (Camire & King, 1991); (iv) the starch present cannot be totally gelatinized in the presence of fibre and is thus not capable of supporting expansion (Camire & King, 1991); and (v) the porous structure of the extrudate depends of the plasticity of the mass before the die, for which starch is mainly responsible. Porosity, defined by the existence of fine pores and a tender structure, is influenced by alterations in the plasticity of the mass, affected by the composition of the mix. Formulations can be enriched by plasticizing substances or by non-plasticizing substances that retard expansion by diluting starch, as is the case of fibres

In the extrusion process of expanded products with low moisture, the expansion of the final product is inversely related to the moisture of the raw material and directly related to the

the faeces interact with the intestinal epithelium (Thebaudin et al., 1997).

that would be more soluble (Fornal et al., 1987; Wang et al., 1993).

(Roberfroid, 1993; Stark & Madar, 1994).

(Colonna et al., 1998).

**3.6 Moisture and temperature**

increase in extrusion temperature; however, the effect of moisture is more significant (Harper, 1994).

Water acts as a plasticizer for the starchy material that displaces itself within the extruder, reducing viscosity and mechanical energy, producing higher density products and inhibiting bubble growth. Studies carried out with corn grits demonstrated that expansion is inversely proportional to the moisture content of the material being extruded (Chinnaswamy, 1993; Colonna et al., 1998). With higher moisture, starch gelatinization is reduced and bubble growth is retarded, resulting in denser and less crunchy final products (Ding et al., 2005).

Inside the extruder, the product that contains molten starch in its composition, when leaving the extruder, has part of its water rapidly evaporated. This water loss is of 3 to 5% and contributes to cooling the product. Subsequent cooling occurs more slowly due to the low thermal conductivity of the extrudate. Also, when emerging from the die, the extrudate undergoes an abrupt pressure fall that also contributes to its expansion (Colonna et al., 1998). The expanded final product presents air cells that are formed due to superheated water vapour pressure. As the temperature of the extrudate is reduced below its glass transition temperature (Tg), it solidifies and maintains its expanded form (Riaz, 2000).

In high moisture extrudates, expansion occurs when the product exits the die, but the structure collapses before the necessary cooling, resulting in a dense and hard product (Harper, 1994).

Another important parameter for extrudate expansion is process temperature. Products do not expand if the temperature does not reach 100°C. Expansion increases with the increase in temperature when moisture content of the material is close to 20%, due to lower viscosity, permitting a more rapid expansion of the molten mass, or due to an increase in water vapour pressure. At low extrusion temperatures, expansion is reduced because starch is not completely molten. Radial expansion degree is proportional to temperature up to a certain value, decreasing at much higher temperatures. The reduction of expansion at very high temperatures is attributed to an increase in dextrinization, weakening starch structure (Colonna et al., 1998).

At high temperatures, the gel is more elastic, forming a matrix with small uniform cells, while at low moisture, the gel formed is not very elastic and the extruded material has large, not very uniform cells. It is expected that an increase in temperature should reduce viscosity of the molten material, favouring bubble growth and producing low density extrudates, with finer cells and greater crunchiness (Ding et al., 2005).

In high moisture extrusion, the properties of the protein extrudates are strongly influenced by extruder conditions (Thiebaud et al., 1996; Noguchi, 1998). Hayashi et al. (1991) reported that extruder barrel temperature was the most important parameter for the texturization of dehulled soybean. Melt temperature is a critical factor in protein cross-linking reactions. Increasing temperature from 140 to 180°C results in a proportional decrease in disulfide linkages formed in extruded soy protein isolates (Areas, 1992). Temperatures lower than 90°C hinder expansion and layer formation (Cheftel et al., 1992). At a given temperature, higher moisture contents result in softer and less texturized extrudates due to reduced

Thermoplastic Extrusion in Food Processing 279

The effects of extrusion cooking on nutritional quality are ambiguous. Benefits include destruction of antinutritional factors, gelatinization of starch, increased soluble dietary fibre and reduction of lipid oxidation. On the other hand, Maillard reactions between protein and sugars reduce the nutritional value of the protein, depending on the raw material types, their composition and process conditions. Besides, heat-labile vitamins may be lost to

Starch digestibility is largely dependent on complete gelatinization. High starch digestibility is essential for specialized nutritional foods such as infant and weaning foods. Creation of resistant starch by extrusion may have value in reduced calorie products (Guy, 2001;

The nutritional value of vegetable proteins is generally enhanced by mild extrusion cooking conditions due to the increase in digestibility (Asp and Björck 1989; Arêas, 1992), probably a result of protein denaturation and the inactivation of enzyme inhibitors present in raw materials, by the exposure of new active sites for enzyme attack (Colonna et al., 1989).

Processing nutritional food products at moisture levels below 20% has been proven to be uneconomical and nutritionally undesirable. Low-moisture extrusion results in production of certain undesirable dextrins as a result of increased shear energy inputs. Losses of vitamins and reduced amino acid availability are greatly accelerated as extrusion moistures are decreased. For this reason, vitamins and heat-sensitive nutrients are usually added post

Mild extrusion conditions (high moisture content, low residence time, low temperature) improve nutritional quality, while high extrusion temperatures (higher than 200°C), low moisture contents (lower than 15%) and/or improper formulation (e.g. presence of highreactive sugars) can affect nutritional quality adversely. Also, to obtain a nutritionally balanced extruded product, careful control of process parameters is essential (Singh et al.,

A benefit derived from extrusion-cooking is the partial or total destruction of potentially antinutritional factors, especially protease inhibitors, haemagglutinins, tannins and phytates, which limit utilization of nutrients in legume seeds. However, chemical alteration produced by thermal treatment could also result in decreased nutrient assimilation,

Vitamin losses in extruded foods vary according to the type of food, moisture content, processing temperature and retention time. Generally, losses are minimal in cold extrusion. The HTST conditions in extrusion cooking, the short residence time of the extrudate and the rapid cooling as the product emerges from the die, cause relatively small losses of vitamins

Extrusion cooking was reported by Saalia & Phillips (2011) as an efficient process to destroy or inactivate aflatoxins, if special conditions (high shear, high temperature, and adequate

Zhu et al. (1996) emphasized that by extrusion cooking, soybeans can be converted into high quality food ingredients. The short residence time and high temperature in an extruder

including lower apparent absorption of certain minerals (Alonso et al., 2001).

and essential amino acids (Fellows, 2000).

extrusion when processing at low moisture conditions (Huber, 2001).

**5. Influence on nutritional quality**

varying extents (Singh et al., 2007).

Riaz, 2000).

2007).

pH) are used.

protein–protein interactions and lower viscosity. At relatively lower moisture contents, higher barrel temperatures (140 to 180°C) result in better textures. At higher moisture levels, temperature needs to be decreased as moisture flash-off may cause considerable water loss if a cooling die is not used (Thiebaud et al., 1996).

High moisture levels combined with elevated temperatures yield extrudates that are very soft and not self-supporting after the die. However, a specially designed die which provides cooling at this section will increase the viscosity of the hot extrudate before exiting, contributing to the correct elasticity and fluidity required for texturization (Noguchi, 1998). The temperature at which solidification occurs is related to the plasticization temperature.

Low moisture (15 to 30%) extrusion tends to result in processes with greater generation of mechanical energy and products with lower density, while high moisture (50 to 70%) extrusion results in products with higher density, and is normally used in pellet production (Guy, 2001).


**8. The spatio-temporal role of the TFG-β pathway in relationship with GDF11 in skin aging**

When stimulating ADSCs through binding to their specific receptor membrane, activin type IIB (ActIIBR), GDF11 activates the SMAD2/3 pathways. Similar mechanism might be achieved on ADSCs by TGF-β, suggesting that interference with TGF-β and GDF11 mechanisms might be the key regulator of healing and aging. The same activated SMAD mechanisms pathway did not achieve similar biological activity and targeted probably different intracellular and intranuclear effectors. These considerations might suggest that even different specific receptors are involved in the extracellular interactions of these factors, a competitive attractivity might be observed between their intracellular effectors, which are the SMAD 2/3 proteins. When the factors compete for these proteins, one can speculate that the favored pathway would be associated either to the amount of the activated receptor (TGF-β or GDF11) or to the sensibility of these receptors to their respective ligands.

Another point of view that could be important to mention is the variation in secretion level of these factors during aging. At this fact, serum level of GDF11 has been reported to decrease during age while TGF-β variation was not really investigated. The antiaging pathway GDF11-dependent could be replaced progressively during aging by the TGF-β regenerating one. This might favor tissue regeneration and repair rather than rejuvenation, leading to the appearance of the symptoms of age.

Thus complicating the dilemma between rejuvenation and regeneration. Secretome composition of ADSCs might be impacted by the aging process or the different epigenetic factors.


**Emanuela Micioni Di Bonaventura 1,**†**, Luca Botticelli 1,**†**, Daniele Tomassoni 2, Seyed Khosrow Tayebati 1, Maria Vittoria Micioni Di Bonaventura 1,\*,**‡ **and Carlo Cifani 1,**‡

Received: 30 September 2020; Accepted: 11 November 2020; Published: 14 November 2020

**Abstract:** The dysfunction of melanocortin signaling has been associated with obesity, given the important role in the regulation of energy homeostasis, food intake, satiety and body weight. In the hypothalamus, the melanocortin-3 receptor (MC3R) and melanocortin-4 receptor (MC4R) contribute to the stability of these processes, but MC3R and MC4R are also localized in the mesolimbic dopamine system, the region that responds to the reinforcing properties of highly palatable food (HPF) and where these two receptors seem to affect food reward and motivation. Loss of function of the MC4R, resulting from genetic mutations, leads to overeating in humans, but to date, a clear understanding of the underlying mechanisms and behaviors that promote overconsumption of caloric foods remains unknown. Moreover, the MC4R demonstrated to be a crucial modulator of the stress response, factor that is known to be strictly related to binge eating behavior. In this review, we will explore the preclinical and clinical studies, and the controversies regarding the involvement of melanocortin system in altered eating patterns, especially binge eating behavior, food reward and motivation.

**Keywords:** melanocortin system; MC3R; MC4R; eating disorders; binge eating disorder; food reward; obesity; *MC4R* mutation; rs17782313; stress


**4. Discussion**

In this study, the computational model of the human placenta and its surroundings has been developed, and the influence of the uterus contraction and elongation on the mechanical environment in the placenta was evaluated by the model. The mechanical load, such as force and stress, in the villous trees and intervillous space, which is corresponding to the placenta region in this model, would be stable although the large and various deformation could be caused by the uterus contraction. As the previous reports have indicated, the characteristic uterus contraction and elongation could deform the placenta [12, 13]. While the structural characteristics of the chorionic

*Histological Properties of the Chorionic and Basal Plates and Maintenance of the Mechanical… DOI: http://dx.doi.org/10.5772/intechopen.112935*

#### **Figure 7.**

*The first principal strain in the computational model of the human placenta and surroundings, caused by the myometrium elongation and contraction (Mode 1, Mode 2, and Mode 3 (***Table 3***)). The distribution of the strain is shown by the entire image (right), and the values along each line, indicated by* **Figure 4** *(left).*

and basal plates and its change along gestational age have been investigated [11], how these plates could influence the mechanical environment on the placenta has been barely examined. It has been reported that the smooth muscle in the marginal zone would actively influence on the uteroplacental vein [18], but its function for the blood circulation in the placenta has been barely examined.

While the thickness of the chorionic plate was just about 1% of the placental thickness [14–16, 21], the chorionic plate is composed of chorion and amnion, whose mechanical characteristics are quite different [11, 21]. Although the thickness and mechanical properties of the basal plate and marginal zone have been barely examined, these parameters in this model were the same as those of the chorionic plate in this model because they connect to the chorionic plate continuously. The results of

this computational model have shown the importance of these thin plates for maintaining the mechanical environment of the intervillous space and villous tree.

The structural characteristics of the chorionic and basal plates and marginal zone has been simplified in this model. For example, amnion could be easily separated at "spongy layer" in the chorionic plate. This behavior would be mechanically important so that this layer will be taken into the computational model in the future. Also, fibrinoid, contained a lot in the placenta, including the chorionic and basal plates, is considered in the model development. Although the mechanical properties have been barely investigated, the estimation based on the previous reports [11, 19, 20] will be helpful.


*4.3.1 Fire cupping*

*Fire cupping* is used with glass, ceramic, and bamboo cups that have no valves as the valve is used to control the air flow through the cup. Traditionally in China, a piece of paper or cotton is used either alighted and inserted into the cup directly, or soaked with 95% alcohol and attached to the end of a stick and then burned. The burning stick is used to make the cup hot and is removed later. This method is associated with a risk of burn [1, 22].


**3. Bone healing inhibitors**

As mentioned above, the process of bone injury regeneration is a complex process that is affected by various factors. In this section, some of the most important negative influencing factors on the bone healing process are mentioned, which should be controlled as much as possible to improve the healing process.

#### **3.1 Non-steroidal anti-inflammatory drugs (NSAID)**

Cyclooxygenase (COX) enzymes classified as COX-1 and COX-2, COX-1 is involved in prostaglandins synthesize in physiological conditions. COX-2 is activated in inflammatory conditions and releases prostaglandins. Whereas, NSAIDs inhibit COX activity [21]. The restriction of prostaglandins synthesized with their critical role in osteoblast and osteoclast functions retarded bone regeneration [22]. However, there is a difference of opinion on the use of NSAID and some studies indicate their positive effects on the treatment process. However, it is better to be careful in using this medicine and limit the duration and dosage [23].