*Article* **Quantum Behavior of a** *PT* **-Symmetric Two-Mode System with Cross-Kerr Nonlinearity**

**Jan Peˇrina Jr. 1,\* and Antonín Lukš 2**

Received: 12 July 2019 ; Accepted: 31 July 2019; Published: 7 August 2019

**Abstract:** Quantum behavior of two oscillator modes, with mutually balanced gain and loss and coupled via linear coupling (including energy conserving as well as energy non-conserving terms) and nonlinear cross-Kerr effect, is investigated. Stationary states are found and their stability analysis is given. Exceptional points for PT -symmetric cases are identified. Quantum dynamics treated by the model of linear operator corrections to a classical solution reveals nonclassical properties of individual modes (squeezing) as well as their entanglement.

**Keywords:** nonlinearly coupled oscillators; PT symmetry; cross-Kerr nonlinearity; stability analysis; quantum properties

#### **1. Introduction**

PT -symmetric systems, which contain gain and loss in mutual balance, have been extensively analyzed in various configurations and from different points of view since the pioneering work by Bender and Boettcher occurred [1–3]. The simplest system is composed of two linearly coupled oscillator modes, one exhibiting gain and the other loss [4]. In real physical applications, there occur additional nonlinear Kerr-type terms in both oscillator modes. They originate in physical models of mode amplification and attenuation typically realized via two-level atoms [5]. These models were developed and extensively discussed in the semiclassical and quantum theories of lasers [6]. Stationary states then occur in such systems due to this nonlinearity. The simplest model of two coupled nonlinear oscillator modes has been generalized to include more oscillators in various configurations. The obtained models were applied in many areas of physics including optical coupled structures [7–10], optical waveguides [11,12], coupled optical micro-resonators [13–15], optical lattices [16–19], opto-mechanical systems [20,21], etc.

Recently, attention has been devoted to the consistent quantum description of PT -symmetric systems. To guarantee the validity of commutation relations among the field operators during the evolution, the fluctuating quantum Langevin forces with specific properties have to be considered in the system [22–25]. As a consequence, the noise in the system constantly increases during the evolution both owing to the amplification and attenuation [24]. Despite this, quantum PT -symmetric systems exhibit interesting and appealing features, such as enhancement of interactions around and at exceptional points (EPs) [26] or quantum Zeno effect [27]. The enhancement of nonlinear interactions then opens the door for the generation of nonclassical light (squeezing) and entangled states [28–31].

Here, we investigate the behavior of a specific form of the model of two coupled oscillator modes with amplification and attenuation that includes only the cross-Kerr nonlinear term. In addition, linear coupling of both modes through *χ*(2) parametric interaction that does not conserve energy, is considered [32,33]. It adds or removes photons simultaneously in both modes. This coupling leads to

nonclassical properties of the fields and the occurrence of entanglement between the modes [22,32,34], together with the cross-Kerr nonlinear coupling [35,36]. The cross-Kerr coupling is known to play a significant role in quantum non-demolition measurement [37], generation of the states defined in finite-dimensional Hilbert spaces [38] and generation of maximally entangled Bell-type states [39,40]). In general, the cross-Kerr coupling appearing in the so-called Kerr couplers considerably changes their quantum properties [41–43]. The cross-Kerr nonlinearity can even enhance the usual Kerr effect, e.g., when squeezing effects are analyzed [44].

We show that continuous sets of stationary states occur in the model and we analyze their stability. Then, in the framework of the model of quantum superposition of signal and noise [22], we address squeezed-state generation and generation of entangled states around the stationary states. We note that if one of the oscillator modes in the analyzed model attains an additional Kerr nonlinear term, only the trivial stationary states exist. On the other hand, if the standard Kerr nonlinear terms are attributed to both oscillator modes, the system behavior considerably changes and only discrete stationary states are found [45]. We note that related systems were analyzed from the point of view of squeezed-state generation in [29] (without parametric interaction) and [31] (no Kerr terms, parametric interaction in individual modes) and quantum-noise generation [25] (without parametric interaction). In addition, the work [46], where breaking of the oscillatory regime in a classical two-mode PT -symmetric system with the Kerr nonlinearity due to larger modes intensities is reported, is worth mentioning.

The paper is organized as follows. In Section 2, the analyzed system is defined and the corresponding Heisenberg equations are given. Stationary states and their stability are investigated in Section 3. Nonclassical properties of the evolving states are discussed in Section 4. Section 5 presents conclusions.

#### **2. Quantum Hamiltonian and Dynamical Equations**

Introducing annihilation (*a*ˆ*j*) and creation (*a*ˆ† *j* ) operators of photons for oscillator modes 1 (*j* = 1) and 2 (*j* = 2) with identical frequencies *ω*, the considered system is described by the following interaction Hamiltonian *H*ˆ [22]:

$$\hat{H} = -i\gamma\_1 \mathfrak{d}\_1^\dagger \mathfrak{d}\_1 - i\gamma\_2 \mathfrak{d}\_2^\dagger \mathfrak{d}\_2 + \left[\epsilon \mathfrak{d}\_1^\dagger \mathfrak{d}\_2 + \kappa \mathfrak{d}\_1 \mathfrak{d}\_2 + \text{h.c.}\right] + \beta\_\mathfrak{c} \mathfrak{d}\_1^\dagger \mathfrak{d}\_2^\dagger \mathfrak{d}\_1 \mathfrak{d}\_2. \tag{1}$$

We assume that mode 1 is attenuated *γ*1 ≥ 0 and mode 2 is amplified *γ*2 ≤ 0. Transfer of energy in the system is described by linear coupling constants  and *κ*. Whereas the coupling constant *-* characterizes transfer of energy between the modes, the constant *κ* quantifies energy inserted and removed to/from both modes in the same amount in the *χ*(2) parametric process. The nonlinear coupling constant *βc* characterizes in Equation (1) the cross-Kerr nonlinear term that is responsible for the occurrence of stationary states. Both *χ*(2) term of parametric interaction and cross-Kerr term occur together in nonlinear photonic structures [33] (waveguides, nonlinear fibers). Symbol h.c. replaces the Hermitian conjugated terms.

The Hamiltonian *H*ˆ in Equation (1) attains its PT -symmetric form provided that the constants *γ*1, *γ*2, *-*, *κ* and *βc* are real and

$$
\gamma\_1 = -\gamma\_2 \equiv \gamma \ge 0. \tag{2}
$$

Moreover, to allow for simple physical interpretation, PT -symmetric Hamiltonians are usually applied for the range of parameters in which their linear parts are endowed with real eigenvalues. For the Hamiltonian *H*ˆ in Equation (1), this occurs provided that

$$
\kappa^2 - \kappa^2 - \gamma^2 \ge 0. \tag{3}
$$

Points in the space of parameters for which equality in Equation (3) holds identify systems with specific properties. They are called exceptional points (EPs) [1]. Without the loss of generality, we further assume *-*> 0 and *κ* ≥ 0.

Applying the canonical commutation relations for field operators [5] we obtain the Heisenberg equations from the Hamiltonian *H*ˆ in Equation (1),

$$\begin{aligned} \frac{d\hat{\mathfrak{a}}\_1}{dt} &= -\gamma\_1 \mathfrak{a}\_1 - i\varepsilon \mathfrak{a}\_2 - i\kappa \mathfrak{a}\_2^\dagger - i\beta\_c \mathfrak{a}\_2^\dagger \mathfrak{a}\_2 \mathfrak{a}\_1 + \hat{l}\_1, \\ \frac{d\hat{\mathfrak{a}}\_2}{dt} &= -\gamma\_2 \mathfrak{a}\_2 - i\varepsilon \mathfrak{a}\_1 - i\kappa \mathfrak{a}\_1^\dagger - i\beta\_c \mathfrak{a}\_1^\dagger \mathfrak{a}\_1 \mathfrak{a}\_2 + \hat{l}\_2. \end{aligned} \tag{4}$$

and the Hermitian-conjugated ones. The fluctuating Langevin operator forces ˆ *l*1 and ˆ *l*2 are introduced in Equation (4) in relation to attenuation in mode 1 and amplification in mode 2, respectively. Their properties [22,23,25],

$$
\begin{aligned}
\langle \hat{l}\_1^\dagger(t) \hat{l}\_1(t') \rangle &= 0, & \langle \hat{l}\_1(t) \hat{l}\_1^\dagger(t') \rangle &= 2\gamma\_1 \delta(t - t'), \\
\langle \hat{l}\_2^\dagger(t) \hat{l}\_2(t') \rangle &= -2\gamma\_2 \delta(t - t'), & \langle \hat{l}\_2(t) \hat{l}\_2^\dagger(t') \rangle &= 0,
\end{aligned}
\tag{5}$$

guarantee validity of the commutation relations for the field operators during the evolution. In mode 1, they express the fluctuation-dissipation theorem [47,48] according to which any dissipation of the energy from a system has to be accompanied by back-action from the environment. In analogy, in mode 2, the Langevin forces represent a part of the 'fluctuation-amplification theorem' that occurs as a consequence of consistent adding the energy into the system [6]. Symbol *δ* means the Dirac function.

#### **3. Stationary States and Their Stability**

First, we address the Heisenberg equations in Equation (4) in their 'classical' noiseless limit, i.e., we write them for the coherent states |*αj* with complex amplitudes *αj* = *j* exp(*iϕj*), *j* = 1, 2:

$$\frac{d\varrho\_1}{dt} = -\gamma\_1 \varrho\_1 + \left[\epsilon \sin(\varrho) - \kappa \sin(\psi)\right] \varrho\_2. \tag{6}$$

$$\frac{d\varrho\_2}{dt} = -\gamma\_2 \varrho\_2 - \left[\epsilon \sin(\varrho) + \kappa \sin(\psi)\right] \varrho\_1. \tag{7}$$

$$\frac{d\varrho\_1}{dt} = -\left[\epsilon\cos(\varrho) + \kappa\cos(\psi)\right]\frac{\varrho\_2}{\varrho\_1} - \beta\_\varepsilon \varrho\_{2'}^2 \tag{8}$$

$$\frac{d\varrho\_2}{dt} = -\left[\epsilon\cos(\varrho) + \kappa\cos(\psi)\right]\frac{\varrho\_1}{\varrho\_2} - \beta\_\varepsilon\varrho\_1^2. \tag{9}$$

In Equations (6) and (7), we suitably replace the phases *ϕ*1 and *ϕ*2 by their sum *ψ* = *ϕ*2 + *ϕ*1 and difference *ϕ* = *ϕ*2 − *ϕ*1.

To reveal the stationary complex amplitudes *α*1 and *α*2, we set the time derivatives in Equations (6)–(9) to zero. Before analyzing the obtained equations in detail, we note that there exist the trivial stationary states with st 1 = st 2 = 0 and arbitrary values of phases *ϕ*st 1 and *ϕ*st 2 . Under the stationary conditions, Equations (8) and (9) for the phases *ϕ*1 and *ϕ*2 are dependent and, e.g., Equation (8) gives us:

$$
\varrho\_1^{\rm st} \varrho\_2^{\rm st} = - (\mathbf{c}\_{\rm c} + \mathbf{c}\_{\rm \kappa}) / \beta\_{\rm c}. \tag{10}
$$

To simplify the notation, we use in Equation (10) and below the following functions that substitute the phases *ψ* and *ϕ*:

$$\mathbf{s}\_{\mathbf{c}} = \boldsymbol{\varepsilon}\sin(\boldsymbol{\varphi}^{\mathrm{st}}), \ \mathbf{c}\_{\mathbf{c}} = \boldsymbol{\varepsilon}\cos(\boldsymbol{\varphi}^{\mathrm{st}}), \ \mathbf{s}\_{\mathbf{x}} = \boldsymbol{\kappa}\sin(\boldsymbol{\psi}^{\mathrm{st}}), \ \mathbf{c}\_{\mathbf{x}} = \boldsymbol{\kappa}\cos(\boldsymbol{\psi}^{\mathrm{st}}).\tag{11}$$

Furthermore, the coupled Equations (6) and (7) considered to be functions of st 1 and st 2 have a nontrivial solution provided that their determinant is zero:

$$
\mathbf{s}\_{\mathbf{c}}^2 - \mathbf{s}\_{\mathbf{c}}^2 + \gamma\_1 \gamma\_2 = 0. \tag{12}
$$

Equation (6) when combined with Equation (10) gives us the stationary solution for amplitudes:

$$
\rho\_{1,2}^{\rm st} = \sqrt{\frac{\gamma\_{2,1}}{\beta\_{\rm c}} \frac{\mathbf{c\_{c}} + \mathbf{c\_{x}}}{\pm \mathbf{s\_{c}} + \mathbf{s\_{x}}}}.\tag{13}
$$

According to Equations (12) and (13), one phase, e.g., *ψ*st in the stationary solution is free. The other phase, *ϕ*st, has to fulfill Equation (12) that admits in general four solutions. The amplitudes st 1,2 determined by Equation (13) have to be real and also Equation (10) has to give nonnegative st 1 st 2 .

To address stability of the stationary solution, we derive the linearized equations for deviations *δ*1, *δ*2, *δϕ* and *δψ* from their corresponding stationary values st 1 , st 2 , *ϕ*st, and *ψ*st:

$$
\frac{d}{dt} \begin{bmatrix} \delta \varrho\_1 \\ \delta \varrho\_2 \\ \delta \varrho \\ \delta \Psi \end{bmatrix} = \begin{bmatrix} -\gamma\_1 & s\_{\mathfrak{c}} - s\_{\mathfrak{k}} & c\_{\mathfrak{c}} \varrho\_2^{\mathfrak{k}} & -c\_{\mathfrak{k}} \varrho\_2^{\mathfrak{k}} \\ -s\_{\mathfrak{c}} - s\_{\mathfrak{k}} & -\gamma\_2 & -c\_{\mathfrak{c}} \varrho\_1^{\mathfrak{k}} & -c\_{\mathfrak{k}} \varrho\_1^{\mathfrak{k}} \\ G^+ & H^+ & I^+ \mathfrak{s}\_{\mathfrak{c}} & I^+ \mathfrak{s}\_{\mathfrak{k}} \\ G^- & H^- & I^- \mathfrak{s}\_{\mathfrak{c}} & I^- \mathfrak{s}\_{\mathfrak{k}} \end{bmatrix} \begin{bmatrix} \delta \varrho\_1 \\ \delta \varrho\_2 \\ \delta \varrho \\ \delta \Psi \end{bmatrix}. \tag{14}
$$

The parameters *G*±, *H*± and *I*± are given as follows:

$$\begin{array}{rcl} G^{\pm} &=& \frac{\beta\_{\mathfrak{c}} \mathfrak{c}\_{1}^{\mathfrak{st}}}{\mathbf{s}\_{\mathfrak{c}} - \mathbf{s}\_{\mathfrak{c}}} \left[ \mathbf{s}\_{\mathfrak{c}} \left( \mp \gamma\_{1} / \gamma\_{2} - 1 \right) + \mathbf{s}\_{\mathfrak{c}} \left( \mp \gamma\_{1} / \gamma\_{2} + 1 \right) \right], \\ H^{\pm} &=& \frac{\beta\_{\mathfrak{c}} \mathfrak{c}\_{2}^{\mathfrak{st}}}{\mathbf{s}\_{\mathfrak{c}} + \mathbf{s}\_{\mathfrak{c}}} \left[ \mathbf{s}\_{\mathfrak{c}} \left( \gamma\_{2} / \gamma\_{1} \pm 1 \right) + \mathbf{s}\_{\mathfrak{c}} \left( -\gamma\_{2} / \gamma\_{1} \pm 1 \right) \right], \\ I^{\pm} &=& \left( \mathbf{s}\_{\mathfrak{c}} - \mathbf{s}\_{\mathfrak{c}} \right) / \gamma\_{1} \pm \left( \mathbf{s}\_{\mathfrak{c}} + \mathbf{s}\_{\mathfrak{c}} \right) / \gamma\_{2}. \end{array} \tag{15}$$

For the PT -symmetric case, eigenvalues *ν* of the dynamical matrix from Equation (14) are given as roots of the following polynomial:

$$\nu \left( \nu^{\natural} + b\nu + c \right) = 0,$$

$$b = 4(\mathbf{c}\_{\mathbf{c}} + \mathbf{c}\_{\mathbf{k}})(\mathbf{c}\_{\mathbf{c}}\mathbf{s}\_{\mathbf{k}}^2 + \mathbf{c}\_{\mathbf{k}}\mathbf{s}\_{\mathbf{c}}^2)/\gamma^2, \ c = -8\mathbf{s}\_{\mathbf{c}}\mathbf{s}\_{\mathbf{k}}(\mathbf{c}\_{\mathbf{c}} + \mathbf{c}\_{\mathbf{k}})^2/\gamma. \tag{16}$$

The eigenvalue *ν*1 = 0 is related to the freedom in determining, e.g., the phase *ψ*st of a stationary state. Provided that c*-* + c*κ* = 0, we have *ν*1−4 = 0 and Equation (13) gives us the trivial stationary state st 1 = st 2 = 0 lying on the border of stability.

Assuming PT -symmetry and special case without *χ*(2) interaction (*κ* = 0), we have s*κ* = c*κ* = 0 and the phase *ψ*st is arbitrary. On the other hand, one solution for the remaining parameters of the stationary state is derived from Equations (12) and (13) for *βc* > 0 in the following implicit form:

$$\mathbf{c}\_{\mathfrak{c}} = \gamma,\ \mathbf{c}\_{\mathfrak{c}} = -\sqrt{\epsilon^2 - \gamma^2},\ \ q\_1^{\mathfrak{st}} = \varrho\_2^{\mathfrak{st}} = \sqrt[4]{\epsilon^2 - \gamma^2} / \sqrt{\beta\_{\mathfrak{c}}}.\tag{17}$$

Similarly, we reveal one stationary solution for *βc* < 0:

$$\mathbf{s}\_{\mathfrak{c}} = \gamma, \; \mathbf{c}\_{\mathfrak{c}} = \sqrt{\epsilon^2 - \gamma^2}, \; \varrho\_1^{\mathfrak{st}} = \varrho\_2^{\mathfrak{st}} = \sqrt[4]{\epsilon^2 - \gamma^2} / \sqrt{-\beta\_{\mathfrak{c}}}.\tag{18}$$

Eigenvalues of the dynamical matrix in Equation (14) are obtained for both solutions in Equations (17) and (18) as *ν*1−4 = 0, i.e., the states are at the border of stability.

For nonzero *κ*, we first address the stationary states in EPs (PT -symmetric case) for specific values of the phase *ψ*st. The trivial stationary states with st 1 = st 2 = 0 and zero eigenvalues *ν*1−4 = 0 in the stability analysis are found under the conditions summarized in Table 1.

**Table 1.** Parameters of the trivial stationary states st 1 = st 2 = 0 with zero eigenvalues *ν*1−4 = 0 for specific values of the phase *ψ*st.

The nontrivial stationary solution with st 1 = st 2 = 2*κ*/*βc* in EPs is revealed for *ψ*st = *π* and *βc* > 0:

$$\mathbf{s}\_{\mathbf{k}} = \mathbf{0}, \ \mathbf{c}\_{\mathbf{k}} = -\mathbf{c}, \ \mathbf{s}\_{\mathbf{c}} = \gamma, \ \mathbf{c}\_{\mathbf{c}} = -\mathbf{c}. \tag{19}$$

On the other hand, we have for *βc* < 0 and *ψ*st = 0 the stationary solution with st 1 = st 2 = 2*κ*/(−*βc*) in EPs provided that

$$\mathbf{s}\_{\mathbf{x}} = \mathbf{0}, \ \mathbf{c}\_{\mathbf{x}} = \kappa, \ \mathbf{s}\_{\mathbf{c}} = \gamma, \ \mathbf{c}\_{\mathbf{c}} = \kappa. \tag{20}$$

Both solutions in Equations (19) and (20) have the same eigenvalues *ν*1,2 = 0 and *ν*3,4 = ±*i*2 √2*κ* in the stability analysis, i.e., no amplification of amplitude fluctuations occur.

For an arbitrary phase *ψ*st, Equation (12) admits four possible values for the phase *ϕ*st:

$$q\mathbf{q}\_1^{\rm st} = q\mathbf{q}\_{\rm base}^{\rm st}, \quad q\mathbf{q}\_2^{\rm st} = \boldsymbol{\pi} - q\mathbf{q}\_{\rm base}^{\rm st}, \quad q\mathbf{q}\_3^{\rm st} = \boldsymbol{\pi} + q\mathbf{q}\_{\rm base}^{\rm st}, \quad q\mathbf{q}\_4^{\rm st} = 2\boldsymbol{\pi} - q\mathbf{q}\_{\rm base}^{\rm st} \tag{21}$$

where *ϕ*st base = arcsin( *κ*2 sin2(*ψ*st) + *γ*2/*-*). However, only some of them lead to real and nonnegative amplitudes st 1 and st 2 in Equation (13) and nonnegative expression in Equation (10). According to Equation (12), |s*-*|≥|s*κ*|. Equation (12) can also be recast into the form suitable for the discussion:

$$
\mathfrak{c}\_{\mathfrak{e}}^{2} - \mathfrak{c}\_{\mathfrak{x}}^{2} = \mathfrak{e}^{2} - \mathfrak{x}^{2} - \gamma^{2}.\tag{22}
$$

Considering Equation (22) for *-*2 − *κ*2 − *γ*2 ≥ 0 we have |c*-*|≥|c*κ*|. If *βc* > 0 [*βc* < 0], Equation (10) requires c*-* ≤ 0 [c*-* ≥ 0] and this admits the phases *ϕ*st 2 and *ϕ*st 3 [*ϕ*st 1 and *ϕ*st 4 ]. The expression (13) for st 1 requires s*-* ≥ 0 independently of the sign of *βc* and so only the phases *ϕ*st 1 and *ϕ*st 2 can be considered. Both conditions are fulfilled only for the phase *ϕ*st 2 [*ϕ*st 1 ] for *βc* > 0 [*βc* < 0].

On the other hand, we have |c*-*|≤|c*κ*| for *-*2 − *κ*2 − *γ*2 ≤ 0. In this case, c*κ* ≤ 0 [c*κ* ≥ 0] is needed in Equation (10) for *βc* > 0 [*βc* < 0] and so *ψ*st ∈*π*/2, 3*π*/2 [*ψ*st ∈0, *π*/2∪3*π*/2, 2*π*]. Similarly as above, s*-* ≥ 0 guarantees nonnegative expression (13) for st 1 for arbitrary *βc*, i.e., only the phases *ϕ*st 1 and *ϕ*st 2 are allowed. According to both conditions, nontrivial stationary states are expected for the phases *ϕ*st 1 and *ϕ*st 2 in the interval of phase *ψ*st ∈*π*/2, 3*π*/2 [*ψ*st ∈0, *π*/2∪3*π*/2, 2*π*] for *βc* > 0 [*βc* < 0].

The EPs occurring at the border of two above discussed regions need special attention. According to Equation (22), we have |c*-*| = |c*κ*| at the EPs. The analysis reveals that on the top of the stationary states characterized in the above two paragraphs, the trivial stationary states with st 1 = st 2 = 0 exist for the phase *ϕ*st 3 in the interval *ψ*st ∈0, *π*/2∪3*π*/2, 2*π* and for the phase *ϕ*st 4 in the interval *ψ*st ∈*π*/2, 3*π*/2 independently of the sign of *βc*.

The above general conclusions are further illustrated in the graphs in Figure 1 where the stationary states and their stability are analyzed in the plane (*γ*/*-*, *ψ*st) for the case *βc* > 0. In Figure 1, we characterize the stationary states by intensities st2 1 and st2 2 of modes 1 and 2, respectively. We judge the stationary states according to the maximal values of imaginary and real parts of the complex frequencies *ν*1−4. A positive (negative) imaginary part means amplification (attenuation) of amplitude fluctuations around the stationary state (we note the inverse notation for signs for *ν* and *γ*). A nonzero real part then indicates oscillations in the evolution of amplitude fluctuations. We have *κ*/*-*= 0.5 for

the graphs drawn in Figure 1 and so the EP occurs for *γ*EP/*-* = √3/2 ≈ 0.87. The stationary solutions for the phase *ϕ*st 1 exist only in the area with exponential increase of amplitudes (*γ* ≥ *γ*EP), they are unstable and amplitude fluctuations oscillate. Only at the EP, the amplitudes st 1 and st 2 are zero and the state is at the border of stability. On the other hand, there exist stationary states for the phase *ϕ*st 2 in the oscillatory regime of amplitude evolution (*γ* < *γ*EP). According to the graph in Figure 1b amplitude fluctuations around these stationary states oscillate. They are amplified except for the line *ψ*st = *π* that lies at the border of stability (see Figure 1d). According to the graphs in Figure 1e–h, the pattern of intensity st2 1 of mode 1 is a mirror image of the pattern of intensity st2 2 of mode 2 with respect to the plane *ψ*st = *π* (compare Equation (13) for *γ*2 = −*γ*1 and ±s*κ*).

The analysis of the graphs for the case with *βc* < 0 drawn under the conditions of the graphs in Figure 1 reveals similarity provided that we replace *ϕ*st 1 by *ϕ*st 2 , shift the phase *ψ*st by *π* and exchange mode amplitudes st 1 and st 2 . This similarity is illustrated in the graphs in Figure 2 where the stationary intensities and stability parameters are drawn at the EP for both cases. Considering *βc* > 0 [*βc* < 0] and following the graphs in Figure 2, there exist only the trivial stationary states with st 1 = st 2 = 0 at the border of stability for *ψ*st ∈0, *π*/2∪3*π*/2, 2*π* [*ψ*st ∈*π*/2, 3*π*/2] and *ϕ*st 2 [*ϕ*st 1 ]. For the remaining phases *ψ*st nontrivial stationary states are found. These states are unstable except for the state with *ψ*st = *π* [*ψ*st = 0] and *ϕ*st 2 [*ϕ*st 1 ] that is at the border of stability and amplitude fluctuations around this state oscillate. Parameters for this state are given in Equation (19) (Equation (20)).

**Figure 1.** Maximal values of the real and imaginary parts of four complex frequencies *ν* in the stability analysis expressed via functions Ω*ν* = log[1 + |Re(*ν*)|] (**a**,**b**) and Γ*ν* = sign[Im(*ν*)] log[1 + |Im(*ν*)|] (**c**,**d**), respectively, and intensities st2 1 (**e**,**f**) and st2 2 (**g**,**h**) of modes 1 and 2, respectively, as they depend on dimensionless attenuation/amplification parameter *γ*/ and phase *ψ*st for stationary states with *ϕ*st 1 (**a**,**c**,**e**,**g**) and *ϕ*st 2 (**b**,**d**,**f**,**h**) defined in Equation (21); symbol sign gives the sign of the argument, log means the decimal logarithm, Re (Im) stands for the real (imaginary) part of the argument, *κ*/*-* = 0.5, and *βc*/*-*= 0.1.

**Figure 2.** Maximal values of the real and imaginary parts of four complex frequencies *ν* in the stability analysis expressed by functions Ω*ν* (**a**,**d**) and Γ*ν* (**b**,**e**), respectively, and intensities st2 of mode 1 (∗) and 2 () (**c**,**f**) as they depend on phase *ψ*st for *ϕ*st 2 (see Equation (21)) and *βc*/*-* = 0.1 (**a**–**c**) and *ϕ*st 1 and *βc*/*-* = −0.1 (**d**–**f**); EP condition *γ* = √*-*2 − *κ*2 is assumed. Functions Ω*ν* and Γ*ν* as well as the other parameters are given in the caption to Figure 1.

#### **4. Quantum Properties of the Evolving States**

We analyze the properties of states evolving from the stationary states determined above and compare them with those characterizing the states originating from non-stationary states. For an initial stationary state, the evolution is described by the Heisenberg equations in Equation (4) linearized around the initial complex amplitudes *α*st 1 and *α*st 2 (*a*ˆ*j* = *α*st *j* + *δa*ˆ*j*, *j* = 1, 2),

$$\begin{array}{rcl} \frac{d\delta\hat{a}\_{1}}{dt} &=& -\left(\gamma\_{1} + i\beta\_{c}|a\_{2}^{\mathrm{st}}|^{2}\right)\delta\mathbb{i}\_{1} - i\left(\boldsymbol{\varepsilon} + \beta\_{c}a\_{1}^{\mathrm{st}}a\_{2}^{\mathrm{st}}\right)\delta\mathbb{i}\_{2} - i\left(\boldsymbol{\kappa} + \beta\_{c}a\_{1}^{\mathrm{st}}a\_{2}^{\mathrm{st}}\right)\delta\mathbb{i}\_{2}^{\dagger} + \hat{l}\_{1} \\ \frac{d\delta\hat{a}\_{2}}{dt} &=& -i\left(\boldsymbol{\varepsilon} + \beta\_{c}a\_{1}^{\mathrm{st}}a\_{2}^{\mathrm{st}}\right)\delta\mathbb{i}\_{1} - i\left(\boldsymbol{\kappa} + \beta\_{c}a\_{1}^{\mathrm{st}}a\_{2}^{\mathrm{st}}\right)\delta\mathbb{i}\_{1}^{\dagger} - \left(\gamma\_{2} + i\beta\_{c}|a\_{1}^{\mathrm{st}}|^{2}\right)\delta\mathbb{i}\_{2} + \hat{l}\_{2}, \end{array} \tag{23}$$

and the Hermitian-conjugated ones. When an initial non-stationary state is assumed, we numerically solve the classical nonlinear Equations (6)–(9) and linearize the Heisenberg equations around the evolving complex amplitudes *α*1(*t*) and *α*2(*t*) [45]. In both cases the solution can be expressed in the following general form

$$
\begin{bmatrix}
\delta\boldsymbol{\delta}\_{1}(t) \\
\delta\boldsymbol{\hat{a}}\_{2}(t) \\
\end{bmatrix} = \mathbf{U}(t) \begin{bmatrix}
\delta\boldsymbol{\delta}\_{1}(0) \\
\delta\boldsymbol{\hat{a}}\_{2}(0) \\
\end{bmatrix} + \mathbf{V}(t) \begin{bmatrix}
\delta\boldsymbol{\delta}\_{1}^{\dagger}(0) \\
\delta\boldsymbol{\hat{a}}\_{2}^{\dagger}(0) \\
\end{bmatrix} + \begin{bmatrix}
\boldsymbol{\hat{f}}\_{1}(t) \\
\boldsymbol{\hat{f}}\_{2}(t) \\
\end{bmatrix},
$$

$$
\begin{bmatrix}
\boldsymbol{\hat{f}}\_{1}(t) \\
\boldsymbol{\hat{f}}\_{2}(t) \\
\end{bmatrix} = \int\_{0}^{t} dt' \mathbf{U}(t - t') \begin{bmatrix}
\boldsymbol{\hat{f}}\_{1}(t') \\
\boldsymbol{\hat{f}}\_{2}(t') \\
\end{bmatrix} + \int\_{0}^{t} dt' \mathbf{V}(t - t') \begin{bmatrix}
\boldsymbol{\hat{f}}\_{1}^{\dagger}(t') \\
\boldsymbol{\hat{f}}\_{2}^{\dagger}(t') \\
\end{bmatrix},
\tag{24}
$$

in which the matrices **U**(*t*) and **V**(*t*) and correlation functions of the fluctuating operator forces ˆ *fj*(*t*), *j* = 1, 2, are determined numerically in general [45] and analytically under specific conditions [35,45].

We consider the initial vacuum state for the evolution of operator amplitude corrections. In this case, the evolving states remain Gaussian and so the following six correlation functions characterize them completely [22,35]:

$$
\langle \delta \mathfrak{d}\_j^\dagger(t) \delta \mathfrak{d}\_j(t) \rangle,\quad \langle [\delta \mathfrak{d}\_j(t)]^2 \rangle,\quad j = 1,2,\quad \langle \delta \mathfrak{d}\_1^\dagger(t) \delta \mathfrak{d}\_2(t) \rangle,\quad \langle \delta \mathfrak{d}\_1(t) \delta \mathfrak{d}\_2(t) \rangle.\tag{25}
$$

They are easily determined from Equation (24). All quantities characterizing the evolving states can then be expressed in terms of the correlation functions (25). For example, the principal squeeze variance of mode *j* is obtained as [49]

$$\lambda\_j = 1 + 2\left[ \langle \delta \hat{a}\_j^\dagger(t) \delta \hat{a}\_j(t) \rangle - |\langle [\delta \hat{a}\_j(t)]^2 \rangle| \right], \quad j = 1, 2. \tag{26}$$

Determination of the covariance matrix in the symmetric operator ordering then allows to reach the logarithmic negativity *EN* that is a suitable quantifier of the entanglement between the modes (for details, see [50,51]).

In Figure 3, we compare the state evolution around a stationary state with that occurring around a non-stationary state. As a stationary state, we consider the state given in Equation (19). The analyzed non-stationary state evolves from the state that differs from that in Equation (19) in the phase *ψ*init = 0:

$$\mathbf{s}\_{\mathbf{x}}^{\text{init}} = 0, \quad \mathbf{c}\_{\mathbf{x}}^{\text{init}} = \mathbf{x}, \; \mathbf{s}\_{\mathbf{c}}^{\text{init}} = -\gamma, \; \mathbf{c}\_{\mathbf{c}}^{\text{init}} = \mathbf{x}. \tag{27}$$

For the stationary state that lies at the border of stability, both intensities 2 1(*t*) and 2 2(*t*) increase during the evolution and the fluctuating forces give dominant contribution to this increase (compare solid and dashed curves in Figure 3a). Contrary to this, for the initial non-stationary state the intensity 2 1(*t*) of attenuated mode 1 first considerably decreases whereas the intensity 2 2(*t*) of amplified mode 2 increases constantly. According to the curves in Figure 3d, the relative contribution of fluctuating forces to the dynamics of intensities is small. In both cases, only the attenuated mode 1 exhibits squeezing (for the principal squeeze variances *λ*1,2, see Figure 3b,e) and both modes are entangled (for the logarithmic negativity *EN*, see Figure 3c,f) for a limited time period. It is worth noting that both squeezing and entanglement are stronger for the initial non-stationary state. The comparison of curves in Figure 3b,d for the principal squeeze variances *λ*1,2 and in Figure 3c,e for the logarithmic negativity *EN* drawn with and without the inclusion of fluctuating forces clearly documents substantial role of these forces in consistent description of PT -symmetric quantum systems.

**Figure 3.** Intensities 2 (**a**,**d**) and principal squeeze variances *λ* of mode 1 (∗) and 2 () (**b**,**e**) and logarithmic negativity *EN* (**c**,**f**) as they evolve along dimensionless time *t* for initial stationary (non-stationary) state with *ψ*st = *π* [*ψ*init = 0], 1,2 = 2*κ*/*βc* and *ϕ*st 2 (**a**–**c**) [*ϕ*init 4 (**c**–**e**)] given in Equation (19) (Equation (27)), assuming *γ* = √*-*2 − *κ*2, *κ*/*-* = 0.5, and *βc*/*-* = 0.1. The initial vacuum state is assumed, evolution is treated with [without] fluctuating forces (black solid [red dashed] curves).

#### **5. Conclusions**

Two oscillator modes with balanced attenuation and amplification were considered to be mutually coupled via the usual linear coupling, *χ*(2) parametric process and cross-Kerr nonlinearity. Nontrivial stationary states that occur owing to the cross-Kerr nonlinearity were identified and their stability was determined. The stationary states typically form one-parameter systems. There occur only unstable stationary states and states lying at the border of stability (zero imaginary parts of frequencies in the stability analysis). The solution of linearized operator equations for mode amplitudes around these stationary states revealed nonclassical properties of the evolving states (single-mode squeezing, entanglement). Initial non-stationary states seem to be more suitable for nonclassical-state generation than the stationary ones at the border of stability. Substantial role of the fluctuating Langevin operator forces in consistent description of the system was demonstrated.

**Author Contributions:** Conceptualization, J.P.J. and A.L.; Investigation, J.P.J. and A.L.; Software, J.P.J.; Writing—original draft, J.P.J.; Writing—review & editing, J.P.J. and A.L.

**Funding:** J.P. acknowledges the support by the GA CR project 18-22102S. A.L. gratefully acknowledges the ˇ support from the project IGA\_PrF\_2019\_008 of Palacký University.

**Acknowledgments:** The authors thank J. K. Kalaga and W. Leo ´nski for discussions about PT-symmetry and Kerr effect.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).


Preface

Mining is an ancient activity that began at the inception of civilization. On re-examining the mining activities of the present day, it is apparent that several transformations in mining techniques have occurred. This book examines the past, present, and future missions of mining. The growing needs of the mining industry have led to larger investments in the field of "mining science," which encompasses engineering, mechanics, chemistry, physics, and management. In the next few decades, "conventional methods" will be replaced by advanced and modern techniques of fast and safe excavations involving technology, machinery, and equipment.

Written by authors from Poland, India, Australia, Brazil, Turkey, Jordan, and Slovakia, this volume includes nine chapters over two sections. Section 1 examines current mining methods and techniques such as open-pit mining and hill mining. Other chapters in this section examine environmental degradation and reclamation due to surface coal mining as well as mining in hills and tunneling.

Section 2 includes chapters on future mining techniques such as shaft excavation, which is examined using a case study of the shaft-sinking experiences of Polish engineers at the Leon IV shaft of the Rydułtowy Mine, Poland.

Further, 'mineral resources' are hidden treasures that can give a boost to commercial activities, the industry as well as entrepreneurs. Activities related to these resources can generate employment and undoubtedly make a significant contribution to GDP growth for mineral-rich states and countries. As far as my optimistic understanding of S&T, R&D and mining system and subject as a whole goes, I believe that this book will be useful for all professionals of the mining industry.

This book is an important resource on mining, minerals, the environment, and engineering. Its applied research concepts and comprehensive content will be beneficial for researchers, students, policy formulators, and decision-makers in different areas of the mining industry. Mine operators and civil or excavation engineering professionals will also find the information presented herein useful and practical.

Compiling and editing a book on the past, present, and future of various mining techniques is a stupendous task, as the mining industry worldwide is extensive and varied. Therefore this book does not cover every aspect of mining; that is a challenge for a future book.

As an editor, I would like to thank those who assisted me with this book project, particularly my organization (CSIR-Central Institute of Mining and Fuel Research) and all my colleagues, of which there are too many to name. I am indebted to them for all their help.

> **Dr. A.K. Soni** Chief Scientist, Mining Technology Group, Nagpur Research Centre, CSIR-Central Institute of Mining and Fuel Research (CSIR-CIMFR), Nagpur, India

> > Section 1

Mining Techniques - Past

and Present

Section 1


7. Optimising cytoreductive surgery

In the United Kingdom, the chief medical officer, Professor Dame Sally Davies recommended that 'the Royal College of Obstetricians and Gynaecologists should make sure that subspecialist training in gynaecological oncology equips doctors to perform optimal surgery for gynaecological cancers and reduce mortality from ovarian cancer' [43].

Both modifiable (technical skills, organisation of care) and constitutional (e.g. age, biology, functional status) factors need to be addressed if we are to make incremental improvements in the outcomes for our patients.

One approach to this could frame any death from ovarian cancer as an 'error'. With such an approach one can adopt a sophisticated approach to addressing the modifiable factors [44, 45]. Vincent and colleagues articulated seven levels at which an error should be tackled. These are institutional (and national climate), organisation & management, work environment, team, individual staff member, task and patient [45]. How this model could be used for maximising optimal cytoreductive surgery is briefly illustrated here.

The statement by the CMO above creates a national sense of the overall aim. In addition, the European Society of Gynaecological Oncology (ESGO) published a benchmark regarding ovarian cancer surgery, which adds to this impetus, about the desirable end goal [46]. This expert consensus report recommends 10 indicators relating to structural, process and outcome metrics ranging from individual performance to team decision making process within a multidisciplinary setting. Such a suite of benchmarks can set the direction of travel.

Around the globe and even within the 'developed' nations the practice of patient selection, planning and delivery of upfront surgery varies a great deal. Therefore in terms of organisation and management, centralization of care delivery will help optimise outcomes [26, 47]. With regard to team and individual factors, training should target both and the end goal should be clear to both entities, that is, microscopic clearance of the disease. The delivery of surgery could be by a sufficiently trained gynaecological oncologist with the appropriate contribution from allied surgical oncologists. ESGO have organised or sponsored numerous workshops targeting technical skills and the philosophy to support primary debulking surgery. Such a concerted effort will help to redress one of the concerns about current training. The application of the current 'gold standard' evidence from two randomised controlled trials, (though heavily criticised) will adversely impact the training of the next generation of surgeons. Further, the timing of TRUST trial outcome and retirement of those trained prior to the wide acceptance of PDS, will change the landscape of practitioners.

Indeed individual and team training can be augmented by not only skills training in workshops, but through buddy operating during the transition phase. Such an endeavour could be supported by virtual platforms to maintain skills (e.g. video based feedback on specific subtasks in cytoreductive surgery) [48, 49]. This will augment task performance.

Patient selection is discussed elsewhere in this book and will not be discussed here. The factors will include demographic, biological (including molecular characteristics of the disease) and radiomics to mention a few [50].

Another important measure in improving the outcomes as part of CRS is to enhance chemotherapy with the use of Hyperthermic Intraperitoneal Chemotherapy (HIPEC). In a recent study, addition of HIPEC to CRS appears to prolong the overall survival by almost 12 months in patients undergoing interval debulking surgery [51]. This offers an exciting complement to maximal effort CRS. These findings will require independent verification prior to widespread adaptation. This topic will not be discussed further in this chapter.


*Edited by Gerhard Kostner and Indumathi Chennamesetty*

In mammalian blood plasma there exist some 6 major lipoprotein classes. Under physiological conditions lipoproteins are certainly beneficial as they transport nutrients and steroids to numerous organs for further metabolism. On the other hand, under pathophysiological conditions most lipoprotein classes promote atherogenesis except of HDL that are considered to be anti-atherogenic. Lipoprotein research is a wide field comprising basic science, analytical methods and clinical investigations. Thus this issue does not raise the claim to give a comprehensive picture of the current knowledge, but rather focuses on specific questions related to animal models in lipoprotein research as well as features of the most atherogenic lipoprotein, Lp(a).

Photo by monsitj / DollarPhoto

Lipoproteins - From Bench to Bedside

Lipoproteins

From Bench to Bedside

*Edited by Gerhard Kostner*

*and Indumathi Chennamesetty*


**9. Conclusions**

Breast cancer is a heterogeneous disease and thus, highly variable across individual patients. This heterogenicity arises not only due to the diversity of genetic and molecular aberrations in primary breast cancer cells but also due to the diversity of cellular populations that inhabit the breast tumor microenvironment. Although IL-6 levels are higher in breast tumors and patient sera, the precise source of this IL-6 remains elusive. Importantly, many breast tumor stromal cells provide a paracrine source of IL-6 for breast cancer cells within the breast tumor microenvironment.  Therefore, ERα-negative breast cancer cells may supply the tumor microenvironment with IL-6 by means of autocrine IL-6 production to exacerbate the poor prognosis associated with this clinical subtype. It will be critical to determine the specific cellular source of breast tumor-associated IL-6 to advance our understanding of this pleiotropic cytokine in breast cancer progression and metastasis. Moreover, this knowledge will facilitate the validation and subsequent clinical utility of current and novel targeted antagonists of the IL-6/STAT3 signaling network in breast cancer.

Fig. 2. Breast cancer cell ERα status dictates paracrine vs. autocrine IL-6 utilization.

### **10. References**

174 Breast Cancer – Focusing Tumor Microenvironment, Stem Cells and Metastasis

mechanisms underlying IL-6/STAT3 signaling in breast cancer cells such as enhanced growth, induction of EMT, multidrug resistance, and recruitment of peripheral fibroblasts. Taken together, accumulating preclinical and clinical data emphasize IL-6 as a highly attractive therapeutic target in breast cancer. It is therefore imperative that more work be done to evaluate current therapeutics and develop novel agents that target IL-6/STAT3

Multiple strategies could be utilized to target the IL-6/STAT3 pathway, but first and most obvious would be anti-IL-6 neutralizing antibodies. One such anti-IL-6 monoclonal antibody is Siltuximab (CNTO 328). The safety and efficacy of Situximab has been demonstrated in preclinical studies and phase I/II clinical trials of diverse human pathologies and malignancies including Castleman's disease (van Rhee *et al.*, 2010), multiple myeloma (Hunsucker *et al.*, 2011; Voorhees *et al.*, 2007), prostate cancer (Cavarretta *et al.*, 2007; Cavarretta *et al.*, 2008; Dorff *et al.*, 2010; Karkera *et al.*, 2011), renal cell carcinoma (Puchalski *et al.*, 2010; Rossi *et al.*, 2010), non-small cell lung cancer (Song *et al.*, 2010), and ovarian cancer (Guo *et al.*, 2010). Furthermore, IL-6R can be targeted with tocilizumab, an anti-IL-6R monoclonal antibody that has shown promising results in IL-6 driven autoimmune diseases (Tanaka et al., 2011) and was recently approved by the FDA for the treatment of rheumatoid arthritis. The promiscuous IL-6 coreceptor, gp130, also has an endogenous soluble form (sgp130) that exclusively inhibits IL-6 *trans*-signaling, thus preserving classical IL-6 signaling. Therapeutic sgp130 would potentially be more targeted toward breast cancer cells, which generally lack membrane-associated IL-6R and therefore utilize IL-6 trans-signaling through IL-6sR. Recombinant soluble gp130 (sgp130- Fc) has been shown to inhibit murine colon carcinogenesis (Becker *et al.*, 2004), suggesting that it may prove effective in breast cancer as well. Finally, a growing number of nonselective kinase inhibitors and recent focus on specific JAK and STAT3 inhibitor development will provide further insight into the roles of JAK and STAT3 in breast

Breast cancer is a heterogeneous disease and thus, highly variable across individual patients. This heterogenicity arises not only due to the diversity of genetic and molecular aberrations in primary breast cancer cells but also due to the diversity of cellular populations that inhabit the breast tumor microenvironment. Although IL-6 levels are higher in breast tumors and patient sera, the precise source of this IL-6 remains elusive. Importantly, many breast tumor stromal cells provide a paracrine source of IL-6 for breast cancer cells within the breast tumor microenvironment.  Therefore, ERα-negative breast cancer cells may supply the tumor microenvironment with IL-6 by means of autocrine IL-6 production to exacerbate the poor prognosis associated with this clinical subtype. It will be critical to determine the specific cellular source of breast tumor-associated IL-6 to advance our understanding of this pleiotropic cytokine in breast cancer progression and metastasis. Moreover, this knowledge will facilitate the validation and subsequent clinical utility of current and novel targeted antagonists of the IL-6/STAT3 signaling network in breast

signaling in breast cancer models.

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cancer.

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**The Role of Fibrin(ogen) in Transendothelial**

Patricia J. Simpson-Haidaris1, Brian J. Rybarczyk2 and Abha Sahni3 *1Department of Medicine, University of Rochester School of Medicine and Dentistry,*

> *2Department of Biology, University of North Carolina at Chapel Hill, 3Aab Cardiovascular Research Institute, University of Rochester*

> > *School of Medicine and Dentistry,*

**Cell Migration During Breast Cancer Metastasis**

Despite all the modern advances in treatment for breast cancer, metastatic disease remains the hurdle to surmount in curing breast cancer or, at least, in significantly reducing morbidity and mortality to improve long-term survival and quality of life. For over a century, inflammation and thrombosis have been linked to metastatic cancer (Boccaccio & Medico, 2006). In addition to being known for describing the factors leading to venous thromboembolism (alterations in blood flow, vascular endothelial injury, and hypercoagulability) as Virchow's triad, in 1863 Virchow noted a connection between chronic inflammation and cancer based on the recruitment of leukocytes to cancerous lesions

Fig. 1. The three faces of cancer metastasis. (Portraits obtained from public domain).

**1. Introduction**

(reviewed in (Balkwill & Mantovani, 2001)) (**Fig. 1**).


**Jazmin Ramirez-Hernandez 1, Ismael Araujo-Vargas 1,\* and Marco Rivera 2**

Academic Editor: Sergio Saponara Received: 9 August 2017; Accepted: 8 September 2017; Published: 12 September 2017

**Abstract:** A study of the fundamental of operation of a three-phase AC-DC power converter that uses Zero-Voltage Transition (ZVT) together with Space Vector Pulse Width Modulation (SVPWM) is presented. The converter is basically an active rectifier divided into two converters: a matrix converter and an H bridge, which transfer energy through a high-frequency transformer, resulting in a modular AC-DC wireless converter appropriate for Plug-in Electric Vehicles (PEVs). The principle of operation of this converter considers high power quality, output regulation and low semiconductor power loss. The circuit operation, idealized waveforms and modulation strategy are explained together with simulation results of a 5 kW design.

**Keywords:** PEV's; AC-DC converter; ZVT; SVPWM; matrix converter; H bridge


**RP in locally advanced PCa: cT3a**

prostatectomy is offered as a radical treatment.

Is defined, as cancer that has perforated the prostate capsule. Surgical treatment has traditionally been discouraged, mainly because patients have an increased risk of positive surgical margins and lymph node metastases and/or distant relapse

In recent years, there has been renewed interest in surgery for locally advanced PCa, and several retrospective case-series have been published. In general, 33.5-66% of patients will have positive section margins, and 7.9-49% will have positive lymph nodes.

On the other hand, excellent 5-, 10- and 15-year overall survival (OS) and cancer-specific survival (CSS) rates have been published.

Therefore, it is increasingly evident that surgery has a place in treating locally advanced disease (45)

As mentioned above, the multimodal treatment is achieving good results, and to corroborate this, several randomised studies of radiotherapy combined with androgen-deprivation therapy (ADT) versus radiotherapy alone have shown a clear advantage for combination

Actually radical prostatectomy is accepted as an election treatment in both low and high risk prostate cancer with different evidence level, and even for very high risk prostate

Gleason score 2-7 and PSA ≤ 20) and a life expectancy > 10 years 1b

Patients with stage T1a disease and a life expectancy >15 yr or Gleason score 7 3

score 8-10 or PSA > 20) 3

score ≤ 8 and a life expectancy > 10 years. 3

The goals of RP are to remove the cancer completely with negative surgical margins, minimal blood loss, no serious perioperative complications, and complete recovery of potency and urinary continence. From an oncologic standpoint, obtaining negative surgical margins is paramount. A positive surgical margin has been associated with as much as 4-fold higher risk of biochemical recurrence, even after adjusting for other prognostic factors such as Gleason grade, extracapsular extension, seminal vesicle invasion, and lymph node metastasis. (48) On the same way, there are good results in terms of morbidity and survival when radical

Is defined, as cancer that has perforated the prostate capsule. Surgical treatment has traditionally been discouraged, mainly because patients have an increased risk of positive

In recent years, there has been renewed interest in surgery for locally advanced PCa, and several retrospective case-series have been published. In general, 33.5-66% of patients will

On the other hand, excellent 5-, 10- and 15-year overall survival (OS) and cancer-specific

Therefore, it is increasingly evident that surgery has a place in treating locally advanced

surgical margins and lymph node metastases and/or distant relapse

have positive section margins, and 7.9-49% will have positive lymph nodes.

INDICATIONS LE

treatment, but no trial has ever proven combined treatment to be superior to RP.

In patients with low and intermediate risk localised PCa (cT1a-T2b and

Optional

Selected patients with low-volume high-risk localised PCa (cT3a or Gleason

Highly selected patients with very high-risk localised PCa (cT3b-T4 N0 or any T N1) in the context of multimodality treatment 3

Optional for selected patients with T3a, PSA < 20 ng/mL, biopsy Gleason

**4. Radical prostatectomy in high risk CaP**

prostatectomy is offered as a radical treatment.

survival (CSS) rates have been published.

**4.2 Evidence of RP in HR.PC RP in locally advanced PCa: cT3a**

**4.1 Introduction**

cancer. (Table 3)

Table 3.

disease (45)

The problem remains on patient selection before surgery: Nomograms, nodal imaging with CT and seminal vesicle imaging with magnetic resonance or directed specific puncture biopsies of the nodes or seminal vesicles can help to identify those patients unlikely to benefit from a surgical approach. (17)

In addition, it is extremely important that radical prostatectomy for clinical T3 cancer requires sufficient surgical expertise to keep an acceptable morbidity level.


*Article* **The E**ff**ect of Tapered Abutments on Marginal Bone Level: A Retrospective Cohort Study**

**Simone Marconcini 1,\*, Enrica Giammarinaro 2, Ugo Covani 1, Eitan Mijiritsky 3, Xavier Vela 4 and Xavier Rodríguez 5**

Received: 8 July 2019; Accepted: 20 August 2019; Published: 24 August 2019

**Abstract:** Background: Early peri-implant bone loss has been associated to long-term implantprosthetic failure. Different technical, surgical, and prosthetic techniques have been introduced to enhance the clinical outcome of dental implants in terms of crestal bone preservation. The aim of the present cohort study was to observe the mean marginal bone level around two-part implants with gingivally tapered abutments one year after loading. Methods: Mean marginal bone levels and change were computed following radiological calibration and linear measurement on standardized radiographs. Results: Twenty patients who met the inclusion criterion of having at least one implant with the tapered prosthetic connection were included in the study. The cumulative implant success rate was 100%, the average bone loss was −0.18 ± 0.72 mm, with the final bone level sitting above the implant platform most of the time (+1.16 ± 0.91 mm). Conclusion: The results of this cohort study suggested that implants with tapered abutments perform successfully one year after loading and that they are associated with excellent marginal bone preservation, thus suggesting that implant-connection macro-geometry might have a crucial role in dictating peri-implant bone levels.

**Keywords:** bone loss; convergence; clinical study

#### **1. Introduction**

Long-term dental implant survival has been extensively documented under different conditions, so that contemporary clinical dentistry has been focusing on means to achieve predictable implant success. Most of the authors agree on the fact that minimal marginal bone loss should be observed one year within the implant loading, as this quantity is a predictor of the long-term implant survival and success [1]. The extent of post-loading bone remodeling has been mainly related to two different phenomena: (1) The microbial infiltration at the implant-abutment (IA) micro-gap—with consequent inflammation and bone demineralization [2]; (2) the implant-abutment (IA) design [3].

The most accounted risk factor for marginal bone loss has been long considered the inflammatory infiltrate at the IA gap [4]. The understanding of the complex biological events impacting the cervical bone surrounding submerged implants begun with the fundamental animal histometric study by Ericsson [5] who typified the inflammatory infiltrate as a consistent finding in matching IA interfaces. This circumscribed inflammation resulted in a round connective demarcation wall that ultimately leads to bone demineralization and resorption [2,6,7]. Different studies indicated less marginal bone resorption around mis-matching implants (implants with a platform switching connection—PS)—when

compared to matching implants—as well as a different organization of the connective tissue fibers [8]. Several theories have been proposed to explain this clinical manifestation, such as the shifting of the inflammatory infiltrate away from the bone, the additional room for protective connective tissue proliferation, or, best, the creation of a geometrical stop for biological width apical establishment. In fact, in matching implants, the fixture first thread is also the first topographic point where the rehabilitation turns from a smaller to a wider diameter, creating a mechanical retention for connective tissues. In short, marginal bone loss should be inevitable, at least to this extent [9]. In PS implants, the implant-abutment discrepancy acts equally, but at a more coronal level—at the platform level—where the connective fibers are retained. It could be hypothesized that the rehabilitation macro-geometry dictates soft and hard tissue position, independent of the effect of the inflammatory infiltrate produced by the gap [10,11].

The gingivally convergent abutment was developed with the idea of maximizing the available space for soft tissues, which is occupied by the bulky metal shoulder in divergent abutments [12]. The sloping profile of gingival convergent abutments would allow tissue to slide coronally in the early phases of healing, creating a thick connective seal above the IA gap.

What is really bearing the brunt of preserving marginal bone levels? Is it either the relative location of the implant-abutment (IA) junction or is it the connection macro-geometry?

The specific aim of this cohort study was to investigate the clinical and radiological outcome of implants with a convergent implant-abutment connection one year after loading.

#### **2. Materials and Methods**

This study was a retrospective, non-interventional analysis of consecutive patients treated with dental implants with a gingivally convergent abutment connection (Shelta XA, Sweden & Martina, Via Veneto 19, 35020, Due Carrare, Padova, Italy). This study was based on patients consecutively treated on a routine basis at one specialistic center (BORG, Carrer de la Mare de Déu de Sales, 67 08840 Viladecans, Barcelona, Spain) during the period from 2016 to 2018.

#### *2.1. Inclusion and Exclusion Criteria*

The medical records of patients who had at least one two-part implant rehabilitated with a convergent abutment with a one-year follow-up were reviewed. Patients were included if presenting a complete set of follow-up radiographs and intra-oral digital photographs. All implants were placed at a slightly sub-crestal level. Patient records were excluded if they did not present for bi-annual follow-up visits, if they had been rehabilitated with overdentures or full-arch prosthesis or if the implants had been placed with simultaneous guided bone regeneration.

#### *2.2. Data Collection and Analysis*

Data were directly entered into an Excel spreadsheet and then converted to a .csv file format in order to be read by the software for statistical analysis. The following population describing the variables were collected: Age, gender, implant characteristics (diameter, length), implant location (tooth number and anterior/posterior, maxillary/ mandibular), type of implant-supported prosthetic restoration (single crown or partial bridge), and follow-up time.

#### *2.3. Radiologic Marginal Bone Level Evaluation*

Routine peri-apical radiographs obtained via the long-cone paralleling technique with a loop film holder (Rinn, Dentsply Australia Pty Ltd, Pacific Hwy, St Leonards NSW 2065, Australia) were used to measure the marginal bone levels. Radiographs were standardized by means of individual resin bites. The distance between the implant–abutment connection and the first bone-to-implant contact (fBIC) on mesial and distal surfaces was recorded. The scale was calibrated by the width of the dental implant achieving a unique pixel/mm ratio (Figure 1). Radiographic bone levels were calculated at the moment of prosthetic transfer connection (impression taking), at loading, and every six months after loading. The mean marginal bone level for each implant was computed merging mesial and distal variations. The marginal bone change was defined as the difference between the last follow-up and the baseline MBL value, with negative values denoting a loss in bone height.

**Figure 1.** The picture is a schematic representation of the calibration performed on the software to achieve bone level linear measurements. The scale was set and calibrated by the width of the dental implant.

All measurements were performed by a single examiner (SM). The intra-examiner reproducibility was evaluated using the intraclass correlation analysis from the measurements in 10 patients, which revealed a strong correlation coefficient of 0.982 for MBL radiological measurements. Measurements were performed via the OsirisX software (Pixmeo SARL, 266 Rue de Bernex, CH-1233 Bernex, Switzerland).


**5. Functional food management**

Country management of functional food is important for effectiveness and safety, but it varies in each country. Japan has been a pioneer in the field of multifunctional food standards. The Ministry of Health, Labor, and Welfare of Japan, for instance, was the first regulatory agency to designate food products as a distinct food section as discussed before in the above topic [22]. The Food for Specified Health Uses (FOSHU) program, which launched in 1991, was the first one to enable health decisions for functional foods backed by scientific data. The Food for Specified Health Uses (FOSHU) seal of approval may be used on the labeling of commodities that have been acknowledged as FOSHU [23]. The volume of Food for Specified Health Uses (FOSHU) approved foods has continuously increased to around 950 as of 2011 [24].

*Categories and Management of Functional Food DOI: http://dx.doi.org/10.5772/intechopen.104664*

#### **Table 3.**

*Cardiovascular disease trained healthcare benefits in terms of dietary element, qualifying authority, region, and claim levels.*

Food is governed in the United States by the Food and drug administration (FDA), Drug, and Cosmetic Act of 1938, which does not include a description of functional foods. This seems to be due to the belief that there are currently sufficient rules regarding the use of food ingredients to cover functional food elements. The main determinant of changes in the laws, as per the Food and Drug Administration (FDA), is the original function of food, and foods in their basic way are regulated. The Nutritional Labeling and Education Act of 1990 covers equally common goods and products for certain health purposes [25].

However, food companies can now make four types of claims on labeling to transmit health messages to consumers. These are some of the subgroups [26]:

On functional labeling requirements, all four kinds of assertions are permitted if they match the established requirements for each benefit to the customers [27]. The FDA webpage has further information concerning the sorts of decisions that can be made on functional foods in the United States. After a rigorous assessment of scientific data submitted to the FDA, food is permitted to bear a health benefit, as per the Nutritional Labeling and Education Act. Extraordinary statements are allowed if there is sufficient scientific consensus or if a single expert of the US government or the National Academy of Sciences issues an acceptable declaration. Whether they may be used on food labels, health claims must always be approved by the FDA. There are now 12 health claims that achieve this important scientific requirement, as well as health claims that are backed up by authoritative statements. The FDA webpage highlights these health decisions [28].

Whenever the statistical backing for a statement has not achieved the greatest degree of scientific proof, trained healthcare claims are used to give details about the diet-disease association [29]. Dermatitis risk, cancer risk, cardiovascular disease risk, cognitive function, diabetes, and hypertension are among the six illness areas for which qualifying health claims are now permitted. **Table 3** lists the dietary components that can be used to make a trained healthcare statement for cardiovascular disease, as well as the statement's particular wording and degree of scientific proof [30].


Improving Accessibility of e-Learning Templates for Students with Disabilities

*Peter Čakš, Monika Ferk, Matjaž Debevc, Julija Bele Lapuh and Ines Kožuh*

## **Abstract**

Because of the effective development of contemporary technologies, also during the recent COVID-19 pandemic, students with disabilities have been increasingly integrated into the society, work, and academic environment. Accordingly, the question of how to provide accessible graphical design of e-learning materials for various categories of students with disabilities and how to comply with international accessibility standards is becoming more and more prominent. The purpose of the study was to propose recommendations for the design of e-learning materials to achieve the preparation of effective study materials for students with hearing and visual loss, as well as autism spectrum disorders. According to their needs and requirements, recommendations for the graphic design of accessible e-learning materials have been designed, developed, implemented, and evaluated. The findings revealed a higher level of acceptance of adapted material for all experimental groups and can be helpful for teachers and other professionals educating students with disabilities.

**Keywords:** e-learning, students with disabilities, accessibility, user interface design

### **1. Introduction**

The spread of information and communication technology (ICT) in all parts of society and the increase of social awareness of the inclusion of persons with disabilities in the social, work, and pedagogical environment has enabled them to be more actively and effectively involved as equal actors in all areas of social activity [1].

Until recently, it was not a common practice for persons with disabilities (PWD) to publicly announce their forms of disability, but this has changed in recent years. For instance, in 2014, 27.1% of people in Europe declared having activity limitations [2]. Moreover, recent findings have revealed that over one billion people live with some form of disability [3]. Likewise, when students with disabilities (SWD) are in question, they are increasingly reporting on specific learning disabilities [4]. Over 10% of students have reported at least one special educational need (SEN), and over 5% have learning disabilities/disorders (LDs) (an overarching group of neurodevelopmental disorders that affect the development of primary and/or secondary academic abilities) [5].

The growing recognition of needs and requirements has also led to the adaptation of the accessibility of various contents, which was also recognized by UNESCO [6]. Assistive technology (AT) is deemed as 'any product whose primary purpose is to maintain or improve an individual's functioning and independence and thereby promote their wellbeing' [7]. Its implementation has the potential to improve functioning, reduce activity limitations, promote social inclusion, and increase participation in education, the labour market and civic life for PWD [8]. To be even more specific, in the study on the impact of AT use for SWD in higher education (HE), the use of AT was substantiated to increase social interactions, provide opportunities for learning support, promote active engagement with peers in course-related discussions, promote engagement in clubs or groups, and encourage the creation of a social group of AT users [8]. Moreover, the gap between assistive and general use technologies in internet tools, software, and hardware has been overcome and has allowed SWD access to a wide range of technologies [9].

Speaking about the learning process for SWD and the use of AT, the concept of e-learning must be considered. The definition of e-learning has changed through the years. Once a new type of training, can nowadays be deemed a web-based learning system for dissemination of information, communication, and knowledge for education and training [10]. E-learning, as a concept, covers a wide range of applications, learning methods, processes, and tools [5] and can be seen as an appropriate tool for SWD. The categorization of e-learning for SWD can be divided into two major groups. The first group refers to the use of AT (hardware or software, used to increase, improve or maintain capabilities of students with disabilities aimed to support and/or increase learning) [11]. The second group of e-learning, which is in the focus of this chapter, refers to a system of procedures, processes, and instructional materials that support learning [5, 12].

Today, computer-based learning management systems are the most used learning systems [13]. The reason is in its very nature. As Moore et al. [14] highlighted, all forms of e-learning could provide learning opportunities for individuals, and the incorporation of technology in education has undoubtedly improved learning for SWD [15]. Therefore, advances in computer-based education are seen as an effective way of remedying this situation by providing assistance and compensation for SWD [16].

The number of SWD attending universities is increasing [17]. Regardless of this fact, many HE institutions do not pay attention to disability, so students continue to face various barriers [9]. These include inaccessible digital course materials and websites, lack of training on how to use needed AT, poor compatibility between software used by the HE institution and students' AT, libraries that do not stock accessible digital textbooks or coursepacks, professors who do not allow students with disabilities to use their mobile technologies in class; procurement of inaccessible HE technologies that will affect teaching and learning for years, the high cost of some AT, and restricted access to computer labs [1]. SWD often report that learning and studying with ICT and related e-learning can help them [18]. It is also generally accepted that through the ICT they can overcome barriers to more successful education [19]. It cannot be overlooked that during recent COVID-19 pandemics, e-learning has been increasingly used to support students' learning processes, especially for SWD [5, 12].

Despite rich guidance and methods for the appropriate design of e-learning materials, which should enable successful accessibility for all, regardless of the type of disability [20, 21], the question arises as to how good graphic design of e-learning is. Do materials also consider the requirements and needs of SWD? The question is

### *Improving Accessibility of e-Learning Templates for Students with Disabilities DOI: http://dx.doi.org/10.5772/intechopen.101931*

also whether there is an effective standardized e-learning course template for SWD. The final question that arises is how the graphic image or template of e-learning material affects the acceptance and effectiveness of the given content according to the type of disability of SWD. While general design guidelines are available [22, 23], there is, to our knowledge, a lack of studies and specific guidelines for the preparation of e-learning templates for SWD concerning different types of disabilities.

In the literature, the studies addressing the requirements of professors, disability service providers, post-secondary graduates, and assistive technologists prevail. At the same time, the problems of designing graphical user interfaces for SWD have still not been sufficiently covered [1]. In their systematic review of papers dealing with the impact of AT on educational and psychosocial outcomes for SWD in higher education, McNicholl et al. [8] found only one article dealing with autism and three with hearing impairments. A few studies have examined some dimensions of e-learning, for instance, learning mathematics [24] and distance learning during COVID-19 [5, 15, 25]. Likewise, Debevc [26] addressed the needs of students with hearing loss where the graphical image of the user interface for a website was developed. It provided the possibility of using a video sign language interpreter in the fixed and pop-up form on the website. However, in this study, there was no research on the acceptance and effectiveness of the user interface template and besides, it was designed for only one SWD category.

According to the literature, no specific standards and guidelines for graphical design of user interfaces for e-learning materials for SWD, in general, could be found, but only general rules for using additional utilities such as captions/subtitles, including an audio description of graphics, providing text-only versions and similar. These general guidelines are also set out in guidelines, such as the web content accessibility guidelines (WCAG), which are also linked to standards such as ETSI EN 301549 "Accessibility requirements for ICT products and services" [27] and other accessibility standards [28].

In the present study, which was part of a larger European Erasmus+ project Trans2Work [29] on transferring students to work environment, we examined whether the graphical design of e-learning course templates affects the acceptance of the given content according to the type of disability of SWD. We focused on the three groups of SWD, which are (1) blind and partially sighted (B&VI), (2) deaf and hardof-hearing (D/HH), and (3) students with autism (only high functional autists with Asperger's syndrome – ASD). We designed, developed, implemented, and evaluated the effectiveness of our customized e-learning template. The study was conducted with the assistance of various national civil associations, NGO representatives, educational institutions, and medical institutions for PWD and SWD.

Based on the findings, we proposed an e-learning template to prepare appropriate e-learning materials adapted for individual types of disabilities in post-secondary education. The primary purpose was to identify the most suitable template for SWD, to design proposals of e-learning materials and to evaluate their efficiency.

## **2. Barriers and solutions in e-learning for students with disabilities**

When preparing e-learning materials for SWD, first and foremost materials should be made accessible. Cooper [30] and Khazanchi [31] point out that e-learning materials should be designed without additional barriers for SWD, and the path to

knowledge should be made easier for them. Every person, regardless of their disability, should have access to information through technology. Through the literature review, we first identified the problems that SWD encounter in the learning process and then sought appropriate solutions.


*Digital Entrepreneurship in Vocational High School Student Level DOI: http://dx.doi.org/10.5772/intechopen.110307*

a breakdown of 16 years totaling 16.7%, students 17-year-olds totaled 41.7%, while 18-year-olds were 41.7%.

From the data that researchers have taken previously, it is known that from twelve respondents of Vocational High School Students the interest in entrepreneurship male students is five students different from female students in number three people with differences who are not too far, while the rest are not interested in doing entrepreneurship is indeed, because there is a difference in motivation between male and female if want to do entrepreneurship as stated by Yuhendri [36] and Adha et al. [37].

Next steps are validity and reliability test. If the value of R is calculated > the value of the table R, then the item on the question instrument is said to be valid. But if the value of R is calculated < R of the table, then the item on the question instrument is said to be invalid and uses a sign of 5% or 0.05. (df = n–2) with data owned by twelve then the value of R table is 0.576 then the value of R counts more than R of the table or it can also be called R count >0.576. As for the reliability test with a significance of 5% or 0.05 that the Cronbach Alpha value > the R value of the table is known to be the Cronbach Alpha value >0.576 then the data is declared Reliable while if the Cronbach Alpha value >0.576 is declared unreliable.

From the **Table 3**, the data of the Digital Entrepreneurship validity test, it can be concluded that (**Table 4**).

So, it can be concluded that the item validity test from digital entrepreneurship is declared valid with the value of r count > r table.

From the reliability test table above, it can be seen that the value of Cronbach's Alpha is 0.966 while the value of the table R is 0.576 so that the research reliability test can be known this is 0.966 > 0.576 then the above reliability data is declared Reliable (**Table 5**).

In the search for results carried out to test the One Sample t Test, we must know that from the data above it is normally distributed so that data testing is carried out as follows (**Table 6**).

The decision basis of the normality test is if the value of Sig. > 0.05 then the data is normally distributed while if the value of Sig. < 0.05 then the data is not normally distributed. So, from the table above, it can be concluded that the Kolmogorov-Smirnov normality test value has a value of 0.078 > 0.05 in other words, the normality test is normally distributed while in the test Shapiro-Wijk normality has a Sig value. by 0.080 > 0.05 then the data is normally distributed. Next, we will do the Independent

#### **Table 4.**

*Conclusions of the digital entrepreneurship validity test.*

#### **Table 5.**

*Digital entrepreneurship reliability test.*

*Source: The data calculated by researchers.*

#### **Table 6.**

*Normality test.*

#### **Table 7.**

*Homogeneity variation test.*

Sample t Test to find out the difference in digital entrepreneurship knowledge in male and female Students, but we must do a homogeneity test as the requirements in the independent t Test analysis, as follows (**Table 7**).

When in homogeneity testing, it is said to contribute homogeneous data if the Significance value >0.05, while if the Significance value is <0.05, the data is not homogeneously distributed. From the data above, it is known that the value of the significance from digital entrepreneurship is 0.106 > 0.05, it can be said that the distribution of the data is homogeneous. Then the requirements to do the Independent Sample t test are met, the independent sample t test is as below (**Table 8**).

In the decision making of the independent sample t test is to find out whether there is an average difference between two unpaired samples, to find out, it can be seen from the value of sig. (2-tailed) < 0.05 then there is a difference between male and female, while if the value of sig. (2-tailed) > 0.05 then there is no difference. Then we see from the table above that the value of sig. (2-tailed) from digital entrepreneurship is 0.026 and 0.036 or it can be mentioned that the value of g. (2-tailed) (0.026) < 0.05 and the value of sig. (2-tailed) (0.036) < 0.05, there is a difference in digital entrepreneurship in male students and female students. These results are supported by research conducted by Adha [37] and Usman [38].

However, schools need to teach digital entrepreneurship to prepare students for business in the 21st century. Several studies support this. Future research on how digital and coding skills and knowledge might help student entrepreneurship could improve