**2. Detection of specific particles by liquid biopsy**

As previously mentioned, in liquid biopsy, circulating tumor cells (CTC), cell-free nucleic acids, exosomes, or tumor-educated platelets (TEPs) can be acquired and studied for different purposes [9, 10].

Circulating tumor cells are cells which detaches from primary tumor and deposits in a patient's blood. The described ability of CTC is particularly important to fully understand metastatic process; therefore a large scientific research field on this topic has evolved. It is believed that CTC can survive in the bloodstream because of the undergoing epithelial-mesenchymal transition (EMT). In this process tumor cells gain plasticity and motility which allows to extravasate from primary tumor into blood and intravasate into distant tissues as well. Detection of CTC can give valuable clinical information about patient's medical status—CTC can be used as a prognostic marker in different carcinomas including malignant thyroid tumors. Analysis of these cells can serve in treatment process as well—response to different pharmaceutical drugs in individual patient can be analyzed [9, 10].

Circulating cell-free nucleic acids (cfDNA, cfRNA, and cfmiRNA) discharge from apoptotic and necrotic tumor cells into the bloodstream. cfDNA originates not only form tumor cells but also from non-tumor cells after exercise, trauma, or inflammation [10]. Extraction of circulating tumor DNA (ctDNA) alone is a difficult process; therefore mutation in cfDNA particles is being searched and indicates the presence of tumor. A lot of mutation has been analyzed in different types of tumors, for example, epidermal growth factor receptor mutation in nonsmall cell lung cancer. At present, it is the only liquid biopsy test that has granted and has FDA approval [9, 11].

By liquid biopsy, cfRNA can be analyzed but this particle is not as stable as cfDNA, therefore it is harder to investigate. Another circulating cell-free nucleic acid—microRNAs—is a new and more stable tumor marker in the blood. Exosomes are microvesicles (40–150 nm) that are released in the blood from tumors and normal cells as well. Exosomes contain proteins, DNA, RNA, miRNA, lipids, and metabolites. Tumor-educated platelets (TEPs) are anucleated cell fragments that can be educated by the transfer of tumor-associated particles, mostly RNA [9, 10].


**Acknowledgements**

This work was funded under U.S. Department of Energy contract number DE-AC07-05ID14517, managed by Battelle Energy Alliance, LLC/Idaho National Laboratory for NASA's Space Nuclear Propulsion (SNP) project in the Space Technology Mission Directorate (STMD).


**1. Introduction**

RNA-seq technology has driven advances in gene expression analysis through new-generation sequencing platforms, as they are versatile, powerful and ensure quality results with accuracy and reproducibility never reached before. This technology generates information that provides meaning to the set of transcripts (transcriptome), opening up possibilities for understanding cell behavior in different environments. RNA is an important component within the cell, since it plays different roles as a messenger regulatory molecule and carrier; and, it is also essential for the maintenance of housekeeping genes [1].

In 2005, the first new generation of sequencing technology was released and has been evolving rapidly [2]. After starting the process of gene expression analysis in bacteria [3, 4] at a more accessible cost, shorter experimental time and without probes, the technology took off and today overlaps other tools used for this purpose, such as microarray technology, until now extremely useful for this type of analysis.

© 2015 The Author(s). Licensee InTech. 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.


Even though land reforms should be country-specific, Moyo et al. (2015) recommends that they clearly define property rights, ensure the security of land tenure and enable land to be used as collateral. For Kanu et al. (2014), the promotion of more equitable land access and rights requires both land registration and legal recognition of customary rights and administrative issues. If they cause land expropriation for smallholder farmers, pastoralists, indigenous communities and other vulnerable groups, land reforms will contribute to food insecurity and increase in poverty and inequality. For instance, land rights and administration have attracted attention in the context of biofuel production and land grabbing by large corporates. In Ghana, Lauretta S. Kemeze, Akwasi Mensah-Bonsu, Irene S. Egyir, D. P. K. Amegashie and Jean Hugues Nlom (Chap. 6) affirm that proper regulation would have avoided the massive conversion of fertile land to biofuel crops, at the expense of food crops. Jatropha cultivation (biofuel sector) could have been promoted on marginal lands so as to not compromise food security. Also, the lack of market for Jatropha seeds highlighted the need to properly regulate the sector in order to protect rural people.

Farmer support organizations are essential in acquiring, applying and continuously disseminating knowledge and skills to farmers (Kanu et al. 2014). Such organizations can help in mitigating risks and increasing investment. Boris Odilon Kounagbè Lokonon (Chap. 8) discusses the extent to which land tenure affects vulnerability to climate shocks, and recommends farmers' labor sharing groups and farmers' organizations. They will lessen vulnerability and raise awareness on relevant technology and good environmental management practices to increase resilience.

Besides, although women constitute most of Africa's labor force in agriculture, rules governing ownership and transfer of land rights are not favorable to women (NEPAD 2013). They face important inequalities in accessing and controlling over land, property and resources. For Kanu et al. (2014), women empowerment is needed in various forms such as policy dialogue, legal reforms, public campaigns, project development, civil society involvement and support to women groups and organizations. Land reforms should also benefit women. Adedoyin Mistura Rufai, Kabir Kayode Salman and Mutiat Bukola Salawu (Chap. 4) plead for the implementation of gender-sensitive policy. According to O. E. Ayinde, T. Abdoulaye, G. A. Olaoye and A. O. Oloyede (Chap. 5), women should be involved in the development and testing of agricultural innovation. If women farmers' preferences are incorporated in the development of agricultural technology, this will ensure food security and increased productivity.

# **13.4 Innovative Financing for the Transformation of African Agriculture**

In sub-Saharan Africa, insufficient cash income and the difficult access to the financial and insurance sectors hampers farmers' ability to develop and to invest. Given the inadequate access to finance, in particular risk capital, subsistence farmers are not able to adopt new varieties and methodologies, hereby constraining agricultural productivity. For Moyo et al. (2015), Africa's general low financial inclusion is even worse in agriculture due to the specific production cycle.2 The challenges of providing acceptable collateral for agricultural lending and adapting loan repayment schedules to crop cycles are huge. As a result, another means for boosting agriculture is to provide actors with the adequate financial resources.

Carren Pindiriri (Chap. 3) recommends the improvement of farmers' access to credit as it has a significant effect on farmers' decision to adopt modern technology. Financial inclusion through the establishment of rural financial institutions can significantly enhance modernization of agriculture. Regarding insurance, Francis H. Kemeze (Chap. 9) discusses the effectiveness of weather index insurance in protecting farmers against climate variability. The results are mitigated due to the specificity of the agriculture sector. Even though insurances cover basis risk, they may not cover the actual on-farm losses. Not only that they do not replace the crop loss but farmers have to buy staple food at increased price in a context of weather shocks. For this reason, the author recommends to complement weather index insurance with supplemental irrigation technology to protect farmers in case of long dry spell or severe drought.

# **13.5 Strengthening Africa's Agriculture Value Chains, Trade and Competitiveness**

A major opportunity for Africa is to build regional value chains in the agro-industry as it will entail agriculture sector growth and job creation. For Kanu et al. (2014), economic potentials for small and medium enterprises are enormous throughout the agricultural value chain, given the untapped agro-industry market opportunity. Greater integration into value chains is expected to boost farmers' benefits and facilitate agribusiness. It will eventually increase trade and integration into global value chains.

Namalguebzanga C. Kafando (Chap. 10) proposes to complete agribusiness development with trade openness and regional integration policies as well as good governance to rapidly benefit from regional and global value chains. It also requires the redefinition of educational policies to efficiently use technology and the development of transport infrastructure. Infrastructure which increases agricultural productivity, reduces post-harvest loss and transportation costs, is particularly in the case of cross-border trade and value chain integration (Kanu et al. 2014). In the same vein, Anani N. Mensah and Abdul-Fahd Fofana (Chap. 12) plead for the improvement of international competitiveness and the support of export companies. There is a need to increase the sophistication and diversification of export products and to move away from forward integration In contrast, African countries should develop backward integration, which means that economies must import primary products from abroad to add value and produce high-tech products locally.

The creation of special economic zones and growth poles has been another means to encourage industrialization and attract foreign direct investment. Joseph Tinarwo (Chap. 10) recommends to integrate SEZ programs into national development strategies so as to ensure high-level political commitment. The author highlights the need to create joint ventures between foreign SEZ companies and local companies with the establishment of low minimum SEZ investment thresholds for local companies. Also, it is important to ensure infrastructure development within and outside the SEZ and to respond to SEZ labor requirements by aligning curricula of universities and Technical Vocational Education and Training (TVET) institutions.

# **13.6 Creative Infrastructure Solutions to Boost and Transform African Agriculture**

Adequate and well-functioning infrastructure is essential for agriculture due to its positive impacts on the costs of delivering agriculture inputs and accessing market for selling outputs. However, sub-Saharan Africa's massive disadvantage in infrastructure (mainly roads, electricity and communications) has increased transaction costs and market risks, especially in small and landlocked countries. The insufficient number of roads has constituted a barrier to agricultural trade and to the adoption of productivity-enhancing inputs. For instance, Christelle Tchamou Meughoyi (Chap. 2) recommends the improvement of infrastructure to supply farmers with the fertilizers they need to adopt improved maize seeds.

Infrastructure is also key to reduce the reliance on rain-fed agriculture and to deal with climate variability through increased irrigation. Idrissa Ouiminga (Chap. 7) highlights the positive effects of soil and water conservation techniques as an alternative to adapting to climate change. Traditional constructed structures or dug are effective to retain water and financially affordable. Zaï (seed holes dug perpendicularly to the slope and staggered), stony ropes or half-moons help farmers to combat land degradation and desertification.

# **Notes**

# **References**

Verdier-Chouchane, A., & Karagueuzian, C. (2016). Moving Towards a Green Productive Agriculture in Africa: The Role of ICTs. *Africa Economic Brief 7*(7). Côte d'Ivoire: African Development Bank.

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/ by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.


**4. Predicting droughts and floods**

In addition to general circulation models (GCMs) [38–40] that attempt to implement physical models of climatic systems, considerable research has been reported over the past decade using machine learning, particularly neural networks, to forecast rainfall [41–48]. The results suggest that the skill of the forecasts using the machine learning approach for medium-term forecast is superior to GCMs [48]. With GCMs, forecasts are usually initially generated for extended grid areas for defined geographical areas. Forecasts for more specific locations can then be generated through a process of downscaling. With the machine learning approach, forecasts are generated for specific locations for which historical data are available. If there are sufficient locations over a geographical region, contour maps representing observed and forecast rainfall can then be generated. This is illustrated for Tasmania in **Figures 2** and **3** for observed and forecast monthly rainfall, 12 months, in advance.

**7**

**Figure 2.**

**Figure 3.**

**5. Rainfall reconstructions**

*Forecast monthly rainfall for Tasmania August 2008.*

degree of certainty, to anthropogenic climate change.

Instrumental records of rainfall and temperature generally extend back only about 100 years in Australia. Reconstructions of past temperatures, extending back hundreds or thousands of years, are available for many parts of the world [49]. These are derived from palaeo data, including tree rings, corals, ice cores and stalagmites. Such reconstructions are much rarer for rainfall and, comparatively, few exist currently for Australia. However, it is important to consider the examples that exist as they enable episodes of droughts and flooding rains to be put into a wider context before asserting particular events are unprecedented, or ascribing them, with a high

Climate proxy data are ideally derived from sources that are located within, or in close proximity to, the region of interest. However, in cases where such proxy

*Introductory Chapter: Australia—A Land of Drought and Flooding Rain*

*DOI: http://dx.doi.org/10.5772/intechopen.89549*

*Observed monthly rainfall for Tasmania August 2008.*

*Introductory Chapter: Australia—A Land of Drought and Flooding Rain DOI: http://dx.doi.org/10.5772/intechopen.89549*

*Rainfall - Extremes, Distribution and Properties*

excessive rainfall.

**3. Flooding rains**

parts of the urban area [3, 33].

repeating 40 year cycle [37].

of herbicides as suggested previously [36].

**4. Predicting droughts and floods**

Australia.

of rainfall in eastern Australia created a greater threat to crop growth than

There are many impacts of droughts other than agriculture [20]. For example, droughts in Australia have had effects on wildlife populations including waterbirds [27, 28]. Studies have shown that droughts have an effect on mental health of the population, particularly in rural areas of Australia [29]. Li et al. investigated the ecological effects of extreme drought [30], including water acidification and eutrophication in the Lower Lakes (Lakes Alexandrina and Albert) in South

Flooding rains are also recurring feature of the Australian climate. For example, prolonged rainfall over large areas of Queensland led to flooding of historic proportions in December 2010, extending into January 2011 [31, 32]. About 33 people died as a result of those floods, with more than 78% of the state (an area larger than France and Germany combined) declared a disaster zone. More than 2.5 million people were affected [31] with approximately 29,000 homes and businesses experiencing some form of inundation, with the cost of flooding estimated to be over A\$5 billion [31]. In January 2011, Brisbane, the state capital of Queensland, experienced its second highest flood in over a century. Major flooding occurred throughout most of the Brisbane River catchment, with an estimated 18,000 properties inundated [32]. More recently in 2018, extreme rainfall conditions inundated the city of Townsville, located in coastal north Queensland, experiencing flooding of large

In addition to the impacts on urban infrastructure, floods may have a substantial impact on the ecosystem. For example, runoff following extreme rainfall has been associated with detrimental impacts on coral of the Great Barrier Reef [34]. Flooding can also impact on the establishment of tree seedlings [35] and has been implicated in the dieback of mangroves in Queensland rivers, rather than the effect

With the devastating impacts of floods, there is interest in understanding and potentially improving predictive capabilities. Studies by McMahon suggest that floods in south-eastern Queensland do not occur randomly but are associated with a

In addition to general circulation models (GCMs) [38–40] that attempt to implement physical models of climatic systems, considerable research has been reported over the past decade using machine learning, particularly neural networks, to forecast rainfall [41–48]. The results suggest that the skill of the forecasts using the machine learning approach for medium-term forecast is superior to GCMs [48]. With GCMs, forecasts are usually initially generated for extended grid areas for defined geographical areas. Forecasts for more specific locations can then be generated through a process of downscaling. With the machine learning approach, forecasts are generated for specific locations for which historical data are available. If there are sufficient locations over a geographical region, contour maps representing observed and forecast rainfall can then be generated. This is illustrated for Tasmania in **Figures 2** and **3** for observed and forecast monthly rainfall, 12 months,

**6**

in advance.

#### **Figure 2.** *Observed monthly rainfall for Tasmania August 2008.*

**Figure 3.** *Forecast monthly rainfall for Tasmania August 2008.*


**2.6 Plasmodium malariae**

*P. malariae* is one of the five human malaria parasites. It causes mild disease, which is therefore called benign malaria. *P. malariae* is found in the Amazon Basin of South America, sub-Saharan Africa, much of southeast Asia, Indonesia, and on many of the islands of the western Pacific [56]. The parasite causes a chronic infection that may sometimes last for a lifetime. Some of the major defining features of *P. malariae* include its longer (72-hour developmental cycle) (quartan periodicity) (compared with the 48-hour cycle of *P. vivax* and *P. falciparum)* and lower parasitemia compared to those in patients infected with *P. falciparum* or *P. vivax* [57, 58]. The signs and symptoms of *P. malariae* malaria include fever, chills and nausea and edema and the nephrotic syndrome has been documented with some *P. malariae* infections [57]. Like that of *P. vivax* and *P. ovale*, Chloroquine is also highly effective against *P. malariae* malaria [55].


**Abstract**

Artificial intelligence (AI) and machine learning (ML) algorithms show promise in revolutionizing many aspects of surgical care. ML algorithms may be used to improve radiologic diagnosis of disease and predict peri-, intra-, and postoperative complications in patients based on their vital signs and other clinical characteristics. Computer vision may improve laparoscopic and minimally invasive surgical education by identifying and tracking the surgeon's movements and providing real-time performance feedback. Eventually, AI and ML may be used to perform operative interventions that were not previously possible (nanosurgery or endoluminal surgery) with the utilization of fully autonomous surgical robots. Overall, AI will impact every surgical subspecialty, and surgeons must be prepared to facilitate the use of this technology to optimize patient care. This chapter will review the applications of AI across different surgical disciplines, the risks and limitations associated with AI and ML, and the role surgeons will play in implementing this technology into their practice.

**Keywords:** artificial intelligence, machine learning, robotics, surgery, nanotechnology, nanosurgery, computer vision, autonomy


**2. Components of functional foods**

Many definitions have been provided for functional foods by different organizations and since it is not legally accepted in any country except Japan, they are not unified. What is identical in all of them is that functional foods are food substances that have been modified in a way so that they have health-related benefits, and may subside the risk of some diseases. They may come from various sources: plants, fruits and vegetables, animals, probiotics, prebiotics, etc. Functional food components can be classified based on the product and the ingredients they contain.


2.3.10. Hydrodynamic Instability

Another relevant feature of the pool-boiling-induced nanoparticle deposition is the hydrodynamic instability alteration [71]. The vapor bubble dynamics during the poolboiling process are affected by the deposition of nanoparticles onto the heating surface, which reduces the distance between the vapor bubble departure points. This distance is commonly designated by the Rayleigh–Taylor wavelength instability wavelength or hydrodynamic instability wavelength, which can be easily identified in the film boiling regime. The alteration of this wavelength is linked to the nanoparticle-deposited layer on the heating surface during boiling, given that the deposited layer will alter the spacing between nucleating bubbles (or vapor columns) and, consequently, the Rayleigh–Taylor instability wavelength. In addition, the hydrodynamic instability approach or hydrodynamic fluid-choking limit connects the characteristic wavelength of the hydrodynamic instability with the CHF enhancement of the nanoparticle-deposited porous layer. The hydrodynamic limit theory is based on the Rayleigh–Taylor instability wavelength and was introduced by Zuber [72] for a plain surface and can be broadened to a coated heating surface having capillary limit. Regarding the hydrodynamic limit, the CHF is generally caused by the vapor columns' instability. The deposited layer can modify the distance between vapor columns on the surface and, consequently, alter the critical instability wavelength. The researchers Liter and Kaviany [73] interpreted the impact of the modulated wavelength on the CHF of a modulated porous layer of nanoparticles (having periodic variations in the layer thickness) using the following expression:

$$Q''\_{prous} = \frac{\pi}{8} \Delta h\_{lv} \sqrt{\frac{\sigma \Delta \rho\_{lv}}{\lambda\_m}}$$

where *λm* is the modulated wavelength or the length scale that defines the vapor escape locations in the working pool-boiling fluid from the porous structure of the heating surface, ∆*hlv* is the enthalpy gradient between liquid and vapor phases, ∆*ρlv* is the density gradient between the liquid and vapor phases, and σ represents the fluid surface tension. For a smooth surface, the *λm* parameter is influenced by the balance between the buoyancy force and surface tension, being a function of the thermal characteristics of the working fluid. In the case of a surface coated with a porous layer, the parameter *λm* depends on the vapor escape pathways and, consequently, is a function of the porous structure of the deposition layer. In the experimental work performed by Park et al. [74], the wavelength alteration corresponded to the employed pool-boiling alumina and graphene/graphene oxide nanofluids CHF amelioration trend. As already stated by Liter and Kaviany, the wavelength can be taken as a well-defined geometrical parameter that depends on the surface conditions. As a consequence, the research team found that the change in the wavelength extends the wetting of the heating surface by enabling the working fluid to break through, resulting in the CHF improvement. Nevertheless, the researchers recognized that the recent published CHF enhancement and wavelengths are not consistent with the prediction of the Liter and Kaviany equation. Accordingly with the findings of the aforementioned authors, Park and Bang [71] stated that the onset of the CHF based on the hydrodynamic limit is due to the instability of vapor columns. The nanoparticle porous layer of the deposited nanoparticles during the boiling process could change the critical distance between vapor columns rising from the heating surface and, consequently, alter the critical instability wavelength. Moreover, a similar situation was observed in the droplet formation on the nanoparticle layer, which is closely linked with the detachment of the bubbles from the heating surface. The authors prepared different nanofluids and the used

pool-boiling apparatus was designed to enable the direct observation of the Rayleigh–Taylor instability wavelengths. The authors reported that the distance between the bubbles was different for each nanoparticle-coated surface, which revealed a shorter distance between the bubbles than that of the bare heating surface. Moreover, the nanofluids that promoted a higher CHF enhancement exhibited shorter Rayleigh–Taylor instability wavelengths. A short wavelength allows the vapor to prevent the formation of a bulk of vapor by venting the vapor evenly across the heating surface. Furthermore, it was demonstrated that the shorter wavelengths also improved the wettability by allowing the liquid to break through the developing vapor film, which also increases the CHF.


**4. Accidental Degeneracy of the Spectrum of** L1

Here, we illustrate the main result, which is absent in the literature so far, to the best of our knowledge. We intend to determine the suitable ladder operators for the degeneracy with respect to the parameter -, which is denoted as accidental degeneracy.

The so-called *accidental degeneracy* of the spectrum of the operator L1 consists of the independence of the eigenvalues *λn* from the parameter -. We explain also this type of degeneracy with the help of ladder operators, denoted by T(±) 2 . Such ladder operators map an eigenfunction *ψn*,-,*m*(**r**) associated with the eigenvalue *λn* either to the null function or to another eigenfunction, denoted by:

$$
\psi\_{n,\ell',m'}(\mathbf{r}) = \mathbb{T}\_2^{(\pm)} \psi\_{n,\ell,m}(\mathbf{r}) .
$$

The two eigenfunctions belong to the same eigenspace of *λn*, that is the value of *n* is the same in both of them, whereas the two values of are different, and the two values of *m* may be either equal or different.

First of all, we establish the conditions for the functions *ψn*,-,*m*(**r**) and T(±) 2 *ψn*,-,*m*(**r**) to be eigenfunctions of L1 associated with the same eigenvalue *λn*. Namely, by virtue of Theorem 1, any ladder operator T(±) 2 has to satisfy the following equality:

$$[\mathcal{Z}, \mathbb{T}\_2^{(\pm)}] = 0.\tag{14}$$

If *g*(*r*) is any function depending on the polar coordinate *r* only, and the operator:

$$
\widetilde{\mathcal{L}}\_{\mathcal{S}} := -\frac{1}{2}\nabla^2 + \mathcal{g}(r),
$$

is defined on the Hilbert space (8), it follows that every operator L1*g* is endowed with rotational invariance. Moreover, there are only two particular circumstances where the operator L1*g* ≡ L1 has a further invariance, which is then "purely accidental" and is responsible for accidental degeneracy. Such cases occur if either *g*(*r*) = *r*2 2 or *g*(*r*) = −1 *r* . The latter case was extensively treated in [3], so we focus on the former case.

A synthetic explanation may sound as follows: We know that the eigenfunction *ψn*,-,*m*(**r**) is associated with the eigenvalues -(- + 1), with respect to which *ψn*,-,*m*(**r**) is also an eigenfunction of the operator *A*(*θ*, *ϕ*) in (7). Furthermore, since the variation of - between two consecutive values is two, this implies that the eigenvalue of *A*(*θ*, *ϕ*), which is subsequent after -(- + 1), is (- + 2)(- + 3). Hence, the raising operator T(+) 2 , whose expression is to be identified, must induce the shift (- + 2)(- + 3) − -(- + 1) = 4- + 6 on the eigenvalues of *A*(*θ*, *ϕ*).

In order to do that, by virtue of Theorem 5, the raising operator T(+) 2 has to satisfy the following condition:

$$[A(\theta,\varphi),\mathbb{T}\_2^{(+)}]\psi\_{n,\overline{l},\mathfrak{M}}(\mathbf{r})=(4\ell+6)\left[\mathbb{T}\_2^{(+)}\psi\_{n,\overline{l},\mathfrak{M}}(\mathbf{r})\right],\tag{15}$$

where *ψn*,¯ -,*m*¯ (*r*) is a particular eigenfunction of *A*(*θ*, *ϕ*). Therefore, we are supposed to identify an operator that verifies both conditions (14) and (15). The underlying degeneracies have different natures. On the one hand, *natural degeneracy* is clarified by the ladder operators T(±) 1 given in (13) and obtained as combinations of the angular momentum operators, and this is due to the fact that the ladder operators have to induce a shift of one unit on the parameter *m*. On the other hand, *accidental degeneracy* has to be clarified by operators T(+) 2 , which are obtained from the combinations of the components of a tensor, because such operators have to induce a shift of two units on the parameter -.

The invariance of the operator L is illustrated by the next result.

**Theorem 7.** *All the components of the following second-rank tensor:*

$$T\_{i\bar{j}} = -\frac{\partial}{\partial r\_i} \frac{\partial}{\partial r\_{\bar{j}}} + r\_i r\_{\bar{j}\prime} \qquad \qquad \text{for } i, j = 1, 2, 3,\tag{16}$$

*where r*1, *r*2, *r*3 *are the coordinates of* **r***, satisfy the commutation identity* [L, *Tij*] = 0*, i.e.,* L *is invariant under the action of all components.*

**Proof.** We can employ the following property of the commutator, which holds for all *A*, *B*, and *C*:

$$[AB, \ C] = A[B, \ C] + [A, \ C]B.$$

Expanding the quantity [L, *Tij*] yields (Some calculations are omitted for the sake of brevity. However, all the calculations are available upon request to the authors.):

$$[\mathcal{L}, T\_{ij}] = \left[ -\frac{1}{2}\nabla^2 + \frac{r^2}{2}, -\frac{\partial}{\partial r\_i}\frac{\partial}{\partial r\_j} + r\_i r\_j \right] = \left[ -\frac{1}{2}\nabla^2, r\_i r\_j \right] + \left[ \frac{r^2}{2}, -\frac{\partial}{\partial r\_i}\frac{\partial}{\partial r\_j} \right]$$

$$= -\frac{1}{2}\sum\_{k=1}^3 \left\{ \left[ -\frac{1}{2}\nabla^2, r\_i r\_j \right] + \left[ r\_{k'}^2 - \frac{\partial}{\partial r\_i}\frac{\partial}{\partial r\_j} \right] \right\}$$

$$= -\frac{1}{2}\left( \frac{\partial}{\partial r\_j} r\_i + \frac{\partial}{\partial r\_i} r\_j + r\_i \frac{\partial}{\partial r\_j} + r\_j \frac{\partial}{\partial r\_i} - r\_j \frac{\partial}{\partial r\_i} - r\_i \frac{\partial}{\partial r\_j} - \frac{\partial}{\partial r\_i} r\_j - \frac{\partial}{\partial r\_j} r\_i \right) = 0,$$

meaning that L is invariant under the action of all nine components *Tij*.

The components *Tij* are the further linear operators that accidentally commute with L, in addition to M1, M2 and M3.

Given the above-mentioned *Tij*, we can consider the following operators:

$$
\mathcal{T}\_1 = T\_{12}, \quad \qquad \qquad \mathcal{T}\_2 = \frac{T\_{22} - T\_{11}}{2},
$$

so that we are able to define the following ladder operators:

$$\mathbb{T}\_2^{(\pm)} := \mathcal{T}\_1 \pm i \mathcal{T}\_2 = -\frac{\partial}{\partial x} \frac{\partial}{\partial y} + xy \pm \frac{i}{2} \left( \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} + y^2 - x^2 \right) \mathcal{T}\_1$$

where *i* = √−1.

**Theorem 8.** *The ladder operators* T(±) 2 *satisfy the following commutation identity:*

$$[\mathbb{M}\_3, \mathbb{T}\_2^{(\pm)}] = \pm 2\mathbb{T}\_2^{(\pm)}.\tag{17a}$$

**Proof.** If we expand the expression of the commutator in the left-hand side of (17a), we obtain:

$$\begin{split} \left[\mathbb{M}\_{3},\mathbb{T}\_{2}^{(\pm)}\right] &= i \left[y\frac{\partial}{\partial x}-\text{x},\frac{\partial}{\partial y},-\frac{\partial}{\partial x}\frac{\partial}{\partial y}+\text{xy}\pm\frac{i}{2}\left(\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}+y^{2}-x^{2}\right)\right] \\ &= -i \left[y\frac{\partial}{\partial x},\frac{\partial}{\partial x}\frac{\partial}{\partial y}\right]+i \left[y\frac{\partial}{\partial x},\text{xy}\right]+i \left[\text{x}\frac{\partial}{\partial y},\frac{\partial}{\partial x}\frac{\partial}{\partial y}\right]-i \left[\text{x}\frac{\partial}{\partial y},\text{xy}\right] \\ &\pm\frac{1}{2}\left\{\left[y\frac{\partial}{\partial x},\frac{\partial}{\partial y}\frac{\partial}{\partial y}\right]+\left[y\frac{\partial}{\partial x},\text{x}^{2}\right]+\left[\text{x}\frac{\partial}{\partial y},\frac{\partial}{\partial x}\frac{\partial}{\partial x}\right]+\left[\text{x}\frac{\partial}{\partial y},\text{y}^{2}\right]\right\} \\ &= i\left(\frac{\partial^{2}}{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}+y^{2}-x^{2}\right)\pm\left(-2\frac{\partial}{\partial x}\frac{\partial}{\partial y}+2xy\right)=\pm 2\text{T}\_{2}^{(\pm)}. \end{split}$$

Using Relations (12a), (12b), (14), and (17a), we can establish the following actions of the operators T(±) 2 on the eigenfunctions *ψn*,-,*m*(**r**):

$$\mathbb{T}\_2^{(+)}\psi\_{n,\ell,m}(\mathbf{r}) = \sum\_{k=0}^{(n-m-2)/2} c\_k \psi\_{n,n-2k,m+2}(\mathbf{r}),\\\mathbb{T}\_2^{(-)}\psi\_{n,\ell,m}(\mathbf{r}) = \sum\_{k=0}^{(n-m+2)/2} c\_k \psi\_{n,n-2k,m-2}(\mathbf{r}).$$

In order to prove that T(±) 2 are the ladder operators that give a clarification of accidental degeneracy, we have to determine the commutators [*A*(*θ*, *ϕ*),T(+) 2 ] and [*A*(*θ*, *ϕ*),T(−) 2 ]. As in Theorem 8, we also obtain the commutators:

[M1, T(+) 2 ] = *i* ' *z ∂ ∂y* − *y ∂ ∂z* , − *∂ ∂x ∂ ∂y* + *xy* + *i* 2 - *∂*2 *∂x*2 − *∂*2 *∂y*2 + *y*2 − *x*2 ( = *i* ' *z ∂ ∂y* , *xy*( + *i* ' *y ∂ ∂z* , *∂ ∂x ∂ ∂y* ( − 1 2 ' *z ∂ ∂y y*2 ( − 1 2 ' *y ∂ ∂z* , *∂*2 *∂y*2 ( = *∂ ∂y ∂ ∂z* − *yz* + *ixz* − *i ∂ ∂x ∂ ∂z* , [M2, T(+) 2 ] = *i* ' *x ∂ ∂z* − *z ∂ ∂x* , − *∂ ∂x ∂ ∂y* + *xy* + *i* 2 - *∂*2 *∂x*2 − *∂*2 *∂y*2 + *y*2 − *x*2 ( = *i* ' *∂ ∂x ∂ ∂y* , *x ∂ ∂z* ( + *i* ' *xy*, *z ∂ ∂x* ( + 1 2 ' *∂*2 *∂x*2 , *x ∂ ∂z* ( + 1 2 ' *x*2, *z ∂ ∂x* ( = *∂ ∂x ∂ ∂z* − *xz* − *iyz* + *i ∂ ∂y ∂ ∂z* ,

from which we can prove the following fundamental result.

**Theorem 9** (**Theorem of accidental degeneracy**)**.** *The commutator A*(*θ*, *ϕ*),T(+) 2 *is the operator:*

$$\mathbb{E}\left[A(\theta,\varphi),\mathbb{T}\_2^{(+)}\right] = 4\mathbb{T}\_2^{(+)}\mathbb{M}\_3 + 6\mathbb{T}\_2^{(+)} + \left(-2i\frac{\partial}{\partial x}\frac{\partial}{\partial z} + 2\frac{\partial}{\partial y}\frac{\partial}{\partial z} + 2ixz - 2yz\right)\mathbb{T}\_1^{(+)} \mathbb{M}\_3$$

*where* T(+) 1 *is the raising operator of the natural degeneracy in* (13)*.*

**Proof.** Using the relation *A*(*θ*, *ϕ*) ≡ M2 1 + M2 2 + M2 3 and expanding the left-hand side, we have:

 *A*(*θ*, *ϕ*), T(+) 2 = M2 1 + M2 2 + M2 3, T(+) 2 = M1 M1, T(+) 2 + M1, T(+) 2 M1 + M2 M2, T(+) 2 + M2, T(+) 2 M2 +M3 M3, T(+) 2 + M3, T(+) 2 M3 = *i* - *z ∂ ∂y* − *y ∂ ∂z* - *∂ ∂y ∂ ∂z* − *yz* + *ixz* − *i ∂ ∂x ∂ ∂z* + *i* - *∂ ∂y ∂ ∂z* − *yz* + *ixz* − *i ∂ ∂x ∂ ∂z z ∂ ∂y* − *y ∂ ∂z* + *i* - *x ∂ ∂z* − *z ∂ ∂x* - *∂ ∂x ∂ ∂z* − *xz* − *iyz* + *i ∂ ∂y ∂ ∂z* + *i* - *∂ ∂x ∂ ∂z* − *xz* − *iyz* + *i ∂ ∂y ∂ ∂z x ∂ ∂z* − *z ∂ ∂x* + 2*i* - *y ∂ ∂x* − *x ∂ ∂y* '− *∂ ∂x ∂ ∂y* + *xy* + *i* 2 - *∂*2 *∂x*2 − *∂*2 *∂y*2 + *y*2 − *x*2 ( + 2*i* ' − *∂ ∂x ∂ ∂y* + *xy* + *i* 2 - *∂*2 *∂x*2 − *∂*2 *∂y*2 + *y*2 − *x*2 (*y ∂ ∂x* − *x ∂ ∂y* .

Expanding the right-hand side yields:

$$4\mathbb{T}\_2^{(+)}\mathbb{M}\_3 + 6\mathbb{T}\_2^{(+)} + \left(-2i\frac{\partial}{\partial x}\frac{\partial}{\partial z} + 2\frac{\partial}{\partial y}\frac{\partial}{\partial z} + 2ixz - 2yz\right)\mathbb{T}\_1^{(+)}$$

$$= 4i\left(-\frac{\partial}{\partial x}\frac{\partial}{\partial y} + xy + \frac{i}{2}\frac{\partial^2}{\partial x^2} - \frac{i}{2}\frac{\partial^2}{\partial y^2} + \frac{iy^2}{2} - \frac{ix^2}{2}\right)\left(y\frac{\partial}{\partial x} - x\frac{\partial}{\partial y}\right)$$

$$+ 6\left(-\frac{\partial}{\partial x}\frac{\partial}{\partial y} + xy + \frac{i}{2}\frac{\partial^2}{\partial x^2} - \frac{i}{2}\frac{\partial^2}{\partial y^2} + \frac{iy^2}{2} - \frac{ix^2}{2}\right)$$

$$+ \left(-2i\frac{\partial}{\partial x}\frac{\partial}{\partial z} + 2\frac{\partial}{\partial y}\frac{\partial}{\partial z} + 2ixz - 2yz\right)\left(iz\frac{\partial}{\partial y} - iy\frac{\partial}{\partial z} - x\frac{\partial}{\partial z} + z\frac{\partial}{\partial z}\right)$$

With the help of some algebra, we can recognize that the two expansions are equal; hence, the proof is complete.

Theorem 9 and the identity T(+) 1 *ψn*-- = 0 lead to establishing the action of the commutator [*A*(*θ*, *ϕ*), T(+) 2 ] on the eigenfunction *ψn*,-,-, obtained when positing *m* = -.

$$\left[A(\theta,\eta)\_{\prime}\operatorname{\mathsf{T}}\_{2}^{(+)}\right]\psi\_{n,\ell,\ell}$$

$$=\left[4\operatorname{\mathsf{T}}\_{2}^{(+)}\operatorname{\mathsf{M}}\_{3} + 6\operatorname{\mathsf{T}}\_{2}^{(+)} + \left(-2i\frac{\partial}{\partial x}\frac{\partial}{\partial z} + 2\frac{\partial}{\partial y}\frac{\partial}{\partial z} + 2ix\operatorname{\mathsf{z}} - 2yz\right)\operatorname{\mathsf{T}}\_{1}^{(+)}\right]\psi\_{n,\ell,\ell}$$

$$=\left[4\operatorname{\mathsf{T}}\_{2}^{(+)}\operatorname{\mathsf{M}}\_{3} + 6\operatorname{\mathsf{T}}\_{2}^{(+)}\right]\psi\_{n,\ell,\ell} + \left(-2i\frac{\partial}{\partial x}\frac{\partial}{\partial z} + 2\frac{\partial}{\partial y}\frac{\partial}{\partial z} + 2ix\operatorname{\mathsf{z}} - 2yz\right)\left[\operatorname{\mathsf{T}}\_{1}^{(+)}\operatorname{\mathsf{V}}\_{n,\ell,\ell}\right]\psi\_{n,\ell,\ell}$$

$$=\left(4\ell + 6\right)\left[\operatorname{\mathsf{T}}\_{2}^{(+)}\psi\_{n,\ell,\ell}\right]\_{\prime}$$

that is we have found the fundamental commutator:

$$\mathbb{E}\left[A(\theta,\varphi),\mathbb{T}\_2^{(+)}\right]\psi\_{n,\ell,\ell} = (4\ell+6)\left[\mathbb{T}\_2^{(+)}\psi\_{n,\ell,\ell}\right].\tag{17b}$$

.

The function T(+) 2 *ψn*,-, is either the null function or a simultaneous eigenfunction of the operator L, with respect to the same eigenvalue *λn* as *ψn*,-,-, and of the operators *A*(*θ*, *ϕ*), M3, with respect to the eigenvalues -(- + 1) + 4- + 6 ≡ (- + 2)(- + 3) and - + 2, respectively, that is:

$$
\mathbb{T}\_2^{(+)} \psi\_{n,\ell,\ell} = \mathbb{C}\_{\ell+2} \,\psi\_{n,\ell+2,\ell+2} \cdot \,\,\,\,
$$

Furthermore, the raising operator T(+) 2 provides a clarification of the accidental degeneracy of the spectrum of the operator L1 because its iterated action on the eigenfunction *ψn*,0,0, where *n* is even, gives:

$$\mathbb{T}\_2^{(+)}\psi\_{n,0,0} = \mathbb{C}\_2\psi\_{n,2,2}, \quad \mathbb{T}\_2^{(+)}\psi\_{n,2,2} = \mathbb{C}\_4\psi\_{n,4,4}, \qquad \dots, \quad \mathbb{T}\_2^{(+)}\psi\_{n,n-2,n-2} = \mathbb{C}\_n\psi\_{n,n,n-2}$$

and analogously with *n* odd:

$$\mathbb{T}\_2^{(+)}\psi\_{n,1,1} = \mathbb{C}\_3\,\psi\_{n,3,3}, \quad \mathbb{T}\_2^{(+)}\psi\_{n,3,3} = \mathbb{C}\_5\,\psi\_{n,5,5}, \qquad \dots, \quad \mathbb{T}\_2^{(+)}\psi\_{n,n-2,n-2} = \mathbb{C}\_n\,\psi\_{n,n,n-1}$$

where the coefficients *Ci* are coefficients of normalization, that is the action of the raising operator T(+) 2 on the eigenfunctions *ψn*,-, leaves the parameters *n* unchanged and modifies the parameters -, *m*, only.

Regarding the operator T(−) 2 , we have the relation (it can be proven by means of the same strategy as in Theorem 9):

$$\mathbb{E}\left[A(\theta,\varphi),\mathbb{T}\_2^{(-)}\right] = -4\mathbb{T}\_2^{(-)}\mathbb{M}\_3 + 6\mathbb{T}\_2^{(-)} + \left(-2i\frac{\partial}{\partial x}\frac{\partial}{\partial z} - 2\frac{\partial}{\partial y}\frac{\partial}{\partial z} + 2ixz + 2yz\right)\mathbb{T}\_1^{(-)}\mathbb{M}\_2$$

where T(−) 1 is the lowering operator of the natural degeneracy in (13), from which we obtain the action:

$$\mathbb{E}\left[A(\theta,\,\,\Psi),\,\mathbb{T}\_2^{(-)}\right]\psi\_{n,\ell,-\ell} = (4\ell+6)\left[\mathbb{T}\_2^{(-)}\psi\_{n,\ell,-\ell}\right].\tag{18}$$

Again, for the above reasons, the function T(−) 2 *ψn*,-,− is either the null function or a simultaneous eigenfunction of the operator L, with respect to the same eigenvalue *λn* as *ψn*,-,−-, and of the operators *A*(*θ*, *ϕ*), M3, with respect to the eigenvalues -(- + 1) + 4- + 6 ≡ (- + 2)(- + 3) and −-− 2, respectively, that is:

$$
\mathbb{T}\_2^{(-)}\psi\_{n,\ell,-\ell} = \hat{\mathcal{C}}\_{\ell+2}\psi\_{n,\ell+2,-\ell-2} -
$$

Since the action of the operator T(−) 2 on the eigenfunctions *ψn*,-,− raises the parameter by two units, as the operator T(+) 2 , we can conclude that there is no lowering operator for the parameter -, but this is not surprising because, by virtue of (17a), the operator T(−) 2 lowers the parameter *m* of the eigenfunctions *ψn*,-,− from − to −- − 2. This means that the parameter cannot change from to -− 2 because otherwise, we would have that:

$$|m| = |-\ell - 2| > \ell - 2,$$

.

which is absurd, due to the constraint |*m*| -

Furthermore, the operator T(−) 2 provides a clarification of the accidental degeneracy of the spectrum of the operator L1 because its iterated action on the eigenfunction *ψn*,0,0, where *n* is even, gives:

$$\mathbb{T}\_2^{(-)}\psi\_{n,0,0} = \breve{\mathbb{C}}\_2\psi\_{n,2,-2}, \ \ \mathbb{T}\_2^{(-)}\psi\_{n,2,-2} = \breve{\mathbb{C}}\_4\psi\_{n,4,-4}, \ \ \dots, \ \ \mathbb{T}\_2^{(-)}\psi\_{n,n-2,-n+2} = \breve{\mathbb{C}}\_n\psi\_{n,n,-n,-2}$$

and also, if *n* is odd:

$$\mathbb{T}\_2^{(-)}\psi\_{n,1,-1} = \breve{\mathbb{C}}\_3\psi\_{n,3,-3}, \ \mathbb{T}\_2^{(-)}\psi\_{n,3,-3} = \breve{\mathbb{C}}\_5\psi\_{n,5,-5}, \ \dots, \ \mathbb{T}\_2^{(-)}\psi\_{n,n-2,-n+2} = \breve{\mathbb{C}}\_n\psi\_{n,n,-n+2}$$

where the coefficients *C*1*i* are coefficients of normalization, i.e., the action of the operator T(−) 2 on the eigenfunctions *ψn*,-,− leaves the parameters *n* unchanged and modifies the parameters -, *m*, only.

#### **5. Discussion**

In this paper, we focused on the accidental degeneracy of a second-order, Schrödingertype differential operator, acting on a Hilbert space. Typically, in the theory of PDEs, the concept of degeneracy is connected to the number of parameters on which the eigenvalues depend. Natural degeneracy and accidental degeneracy were reformulated and characterized by using the ladder operators. Such a useful tool can be further employed to provide a new way to describe degeneracy in eigenvalue problems with elliptic operators.

**Author Contributions:** Methodology, S.P.; Writing—original draft, A.P.; Writing—review—editing, R.D.M. All the authors equally contributed to the present work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study, nor in the writing of the manuscript.


**5. Results**

The results presented in this section highlight the promising potential of the material extracted from silica in raising the pH levels of water and suggest that further research in this area is warranted.


*3.5.1. Total RNA extraction and reverse transcription*

Approximately 30mg of macro-dissected caecal tonsil tissue per sample (previously submerged in RNAlater® (Sigma-Aldrich, USA) and stored frozen at -80°C) was homogenised in 500 uL QIAzol lysis reagent for 10 min at 30 Hz in a Tissuelyzer LT (Qiagen, UK). Lysates were mixed with 100 μL chloroform, transferred to pegGold PhaseTrap tubes (PeqLab, UK) and centrifuged for 5 mins at room temperature. The aqueous phase was poured into fresh tubes, mixed with 1.5 volumes of ethanol and applied to Qiagen RNeasy columns (Qiagen, UK). RNA was purified according to the manufacturer's instructions (Qiagen, UK). RNA integrity was assessed using an Agilent Bioanalyzer and RIN was >8 for all samples. Purity and quantity were measured using a NanoDrop spectrophotometer; for all samples the absorbance peak was at 260 nm, A260/280 > 2 and A260/230 > 1. About 800 ng of RNA were reverse transcribed using a Quantitect reverse transcription kit (Qiagen, UK) in a 10 μL reaction according to the manufacturer's instructions. This RT kit includes a mandatory gDNA wipe out step. The completed reaction was diluted 10-fold with 5 μg/mL tRNA in water.

for product specificity (single peak) and the presence of primer dimers. All primers were designed to be intron-spanning so that any residual gDNA present could not be detected and avoided known SNP and secondary structure. Assays were designed by qStandard (www. qstandard.co.uk) and were tested for specificity by electrophoresis, efficiency >95%, sensitivity to 10 copies/rxn, and linearity over 7 log by qPCR. Copy numbers/reaction were derived from the standard curves using the Rotor-Gene software. The four reference genes identified as the most stable using geNorm software were B2M, GAPDH, PPIA and YWHAZ. The normalisation factor for each sample was used to normalise GOI copy numbers per reaction.

Plant Extracts, Energy, and Immune Modulation in Broilers

http://dx.doi.org/10.5772/intechopen.77220

111

Data were statistically analysed by two way analysis of variance (ANOVA) using a 2 × 2 factorial arrangement of treatments, blocked by experiment. The main effects were the cereals (maize and wheat) and additives (with and without PE). All data were processed using the procedure of Genstat (18th Edition) statistical software (IACR, Rothamstead, Hertfordshire, UK). In all instances, differences were reported as significant at P < 0.05. Graphics (**Figure 2**)

Dietary PE supplementation significantly improved (P < 0.05) gain to feed (G:F) ratio by 2 points and dietary NE by 0.34 MJ (**Table 2**). No changes (P > 0.05) were observed in dietary ME due to PE supplementation. The increase in feed efficiency is in agreement with the ability of spices and mixtures of spices to increase bile secretion, activity of the pancreatic, and brush border enzymes [43, 44]. Maize based diets produced higher (P < 0.05) daily FI and ME, although wheat based diets had higher NE (P < 0.001). The values of ME and NE were in similar to previous reports [9, 45]. In agreement with [24], there were dietary type x PE interactions (P < 0.05) observed in bird growth, as birds fed wheat diets did not respond (P > 0.05) to PE supplementation. Similar tendency (P = 0.074) was observed for daily feed intake. Compared to maize, wheat contains more water-soluble non-starch polysaccharide (NSP), a carbohydrate complex possessing antinutrient activity, which may reduce dietary nutrient availability [46], thus explaining the reduced performance of birds fed wheat based diets. The observed interaction may also be due to the relatively high fat content of the wheat compared to maize based diets, and not to the cereals alone. Widening the dietary ME to protein ratio is likely to affect body fat retention more than bird growth performance, suggesting an explanation for the inconsistency between weight gain and NE of birds fed wheat based diets. However, the impact of dietary formulation (cereals, protein sources, fat content etc.) on the effectiveness of supplementary PE in poultry nutrition warrants further investigation. Although there is a lack of consistency between growth performance and dietary ME, this is in agreement with many studies [8, 10, 11] but is in disagreement with others [9]. The

were produced in "ggplot2" package version 2.2.1. [41] using R version 3.4.1. [42].

**4. Effect of PE on bird growth performance and dietary available**

**3.6. Statistical analysis of data**

**energy**

#### *3.5.2. Quantitative real-time PCR*

Two microlitres of cDNA were amplified in a 10 μL reaction using Agilent Brilliant III SYBR Ultra-Fast SYBR Green mix with each primer at a final concentration of 500 nmol/L. The notemplate control reaction contained 2 μL of tRNA (0.5 μg/mL). DNA standards (10^7–10^1 copies/rxn) for each gene were included in each run. Reactions were pipetted robotically using a Qiagility (Qiagen, UK). Amplification parameters were: 95°C for 3 min followed by 40 cycles of 95°C for 5 sec, 57°C for 1 sec in a Rotor-Gene 6000. Melt curves were checked

**Figure 2.** The effect of dietary plant extracts (PE) on the normalised mRNA copy number (per reaction) of (a) CD40 LG, (b) IL-12B, (c) INFG, (d) IL-6 in chicken caecal tonsils. Error bars represent ±1 pooled SEM.

for product specificity (single peak) and the presence of primer dimers. All primers were designed to be intron-spanning so that any residual gDNA present could not be detected and avoided known SNP and secondary structure. Assays were designed by qStandard (www. qstandard.co.uk) and were tested for specificity by electrophoresis, efficiency >95%, sensitivity to 10 copies/rxn, and linearity over 7 log by qPCR. Copy numbers/reaction were derived from the standard curves using the Rotor-Gene software. The four reference genes identified as the most stable using geNorm software were B2M, GAPDH, PPIA and YWHAZ. The normalisation factor for each sample was used to normalise GOI copy numbers per reaction.