Preface

Veterinary medicine is advancing at a very rapid pace, particularly given the breadth of the discipline. This book examines new developments covering a wide range of issues from health and welfare in livestock, pets, and wild animals to public health supervision and biomedical research.

As well as containing reviews offering fresh insight into specific issues, this book includes a selection of scientific articles which help to chart the advance of this science. The book is divided into several sections. The opening chapters cover the veterinary profession and veterinary science in general, while later chapters look at specific aspects of applied veterinary medicine in pets and in livestock. Finally, research papers are grouped by specialisms with a view to exploring progress in areas such as organ transplantation, therapeutic use of natural substances, and the use of new diagnostic techniques for disease control.

This book was produced during World Veterinary Year 2011, which marked the 250th anniversary of the veterinary profession. It provides a fittingly concise and enjoyable overview of the whole science of veterinary medicine.

> **Carlos C. Perez-Marin**  Department of Animal Medicine and Surgery, Faculty of Veterinary Medicine, University of Cordoba, Spain

**Part 1**

**Veterinary Medicine: General Aspects**


**Yu Huang 1,2,\* and Xu Han 1**

Received: 31 March 2020; Accepted: 27 April 2020; Published: 29 April 2020

**Abstract:** With the accelerated development of marine engineering, a growing number of marine structures are being constructed (e.g., seabed pipelines, drilling platforms, oil platforms, wind turbines). However, seismic field investigations over recent decades have shown that many marine structures were damaged or destroyed due to liquefaction. Seismic liquefaction in marine engineering can have huge financial repercussions as well as a devastating effect on the marine environment, which merits our great attention. As the effects of seawater and the gas component in the seabed layers are not negligible, the seabed soil layers are more prone to liquefaction than onshore soil layers, and the liquefied area may be larger than when liquefaction occurs on land. To mitigate the impact of liquefaction events on marine engineering structures, some novel liquefaction-resistant marine structures have been proposed in recent years. This paper reviews the features of earthquake-induced liquefaction and the mitigation strategies for marine structures to meet the future requirements of marine engineering.

**Keywords:** marine engineering; seismic liquefaction; novel liquefaction-resistant structures; mitigation strategies


*Article* **Exploration of Multiple Transfer Phenomena within Viscous Fluid Flows over a Curved Stretching Sheet in the Co-Existence of Gyrotactic Micro-Organisms and Tiny Particles**

**Pachiyappan Ragupathi 1,†, N. Ameer Ahammad 2, Abderrahim Wakif 3, Nehad Ali Shah 4,† and Yongseok Jeon 5,\***

**Abstract:** In the present study, the magnetohydrodynamics (MHD) bio-convective flow and heat transfer of nanofluid, due to the swimming of the gyrotactic micro-organisms over a curved stretched sheet, is examined. In addition, thermophoresis and Brownian motion behaviors are also investigated by assuming slip conditions at the boundary. A non-linear system of partial differential equations (PDEs) is reduced to a system of ordinary differential equations (ODEs). For convergent solutions, the obtained ODE system is solved by the use of the BVP4C routine integrated MATLAB package. In addition, the impacts of different influential parameters on motile micro-organisms, temperature, velocity, and concentration profiles are deliberated. The velocity field is observed to be reduced when the slip parameter increases. As the main results, it is demonstrated that the distribution of motile microorganisms against the curvature parameter decreases significantly. Similarly, it is found that the nanofluid parameters (i.e., Brownian motion and thermophoresis parameters) and the Peclet number reduce the motile micro-organisms' number. On the other hand, it is evidenced that the motile micro-organisms' distribution can be improved with an increase in bio-convective Schmidt number.

**Keywords:** bio-convection; gyrotactic micro-organisms; curved stretching sheet; slip condition; thermophoresis; Brownian diffusion

**MSC:** 76D05; 82D80


**4.5 Intracranial aneurysms**

Intracranial aneurysms (IAs) are commonplace in the population, with a global estimated prevalence between 2 and 5% [78]. Although most of these aneurysms are asymptomatic, they carry the risk of rupture which if realized leads to a subarachnoid hemorrhage – a prognosis producing a dramatic case fatality of 50% [79]. Thus, there is great interest in the rapid and accurate identification of unruptured intracranial aneurysms on brain imaging.

At the present moment, intra-arterial digital subtraction angiography (IADSA) is the gold-standard for the diagnosis of intracranial aneurysms, with computed tomography angiography (CTA), magnetic resonance angiography (MRA), and transcranial Doppler sonography also shown to be effective diagnostic tests [80]. Time-of-flight MR angiography (TOF-MRA) is a non-invasive, non-contrast enhanced technique that enables discrimination between vessels and stationary tissues by inducing blood inflow effects [81]. Due to the absence of ionizing radiation or intravenous contrast agents, time-offlight MR angiography (TOF-MRA) is typically the first modality of choice for aneurysm screening. Hence, many inroads for DL applications have been explored in this area.

Nakao et al. developed a computer-assisted detection (CAD) deep CNN architecture combined with a maximum intensity projection (MIP) algorithm trained on 450 patients worth of TOF-MRA scans. The team achieved a high sensitivity of 94.2% (98/104) and only 2.9 false positives per case [82]. Faron et al. similarly developed a CNN model finding an overall sensitivity of 90% with a false positive rate of 6.1%. More consequently, the Faron team further found that there was no significant difference in aneurysm detection performance between the CNN model and two blinded diagnostic neuroradiologists, with an overall increase in human detection sensitivity when combining their detection hits with the CNN model's hits (reader 1: 98% vs. 95%, P = 0.280; reader 2: 97% vs. 94%, P = 0.333) [83].

Ueda et al. developed a ResNet architecture algorithm fed with 683 TOF-MRA patient scans and achieved a sensitivity of 91% (592 of 649) and 93% (74 of 80) for their internal and external data sets, respectively [84]. More interestingly, the model improved aneurysm detection in their retrospectively collected TOF-MRA scans by 4.8% (31 of 649) and 13% (10 of 80), respectively, compared to the initial radiologistinterpreted assessments.

Until recently, machine-learning algorithms largely focused on MRA imaging. However, more recent efforts were expanded to include CT-based imaging


**HEPATITIS C - FROM INFECTION TO CURE**

#### **Hepatitis C - From Infection to Cure**

http://dx.doi.org/10.5772/intechopen.72268 Edited by Imran Shahid

#### **Contributors**

Petra Surlin, Ion Rogoveanu, Liliana Foia, Dorin Gheorghe, Dan Nicolae Florescu, Amelia Surdu, Sorina Mihaela Solomon, Vasilica Toma, Livia Melo Villar, Leticia Scalioni, Cristiane A. Villela-Nogueira, Allan P Da Silva, Sidra Rehman, Abdel Rahman Zekri, Bahnassy, Mona S. Abdellateif, Gulam Waris, Akshaya Ramachandran, Binod Kumar, Thomas I Michalak, Snežana Jovanović-Ćupić, Milena Krajnovic, Ana Bozovic, Nina Petrovic, Moustafa Elemeery, Hien Tran, Philip Aston, Alex Cassenote, Cassia J. Mendes-Correa, Katie Cranfield, Haley O'Farrell, Phuong Hoang, George Lankford, Aluisio Segurado, Imran Shahid

#### **© The Editor(s) and the Author(s) 2018**

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First published in London, United Kingdom, 2018 by IntechOpen eBook (PDF) Published by IntechOpen, 2019 IntechOpen is the global imprint of INTECHOPEN LIMITED, registered in England and Wales, registration number: 11086078, The Shard, 25th floor, 32 London Bridge Street London, SE19SG – United Kingdom Printed in Croatia

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Combinatoral Topology and Descompoibilities to Shellability

#### **Chapter 6**

## Vertex Decomposability of Path Complexes and Stanley' s Conjectures

*Seyed Mohammad Ajdani and Francisco Bulnes*

#### **Abstract**

Monomials are the link between commutative algebra and combinatorics. With a simplicial complex Δ, one can associate two square-free monomial ideals: the Stanley-Reisner ideal *I*Δ whose generators correspond to the non-face of Δ, or the facet ideal *I*(Δ) that is a generalization of edge ideals of graphs and whose generators correspond to the facets of Δ. The facet ideal of a simplicial complex was first introduced by Faridi in 2002. Let *G* be a simple graph. The edge ideal *I*(*G*) of a graph *G* was first considered by R. Villarreal in 1990. He studied algebraic properties of *I*(*G*) using a combinatorial language of *G*. In combinatorial commutative algebra, one can attach a monomial ideal to a combinatorial object. Then, algebraic properties of this ideal are studied using combinatorial properties of combinatorial object. One of interesting problems in combinatorial commutative algebra is the Stanley's conjectures. The Stanley's conjectures are studied by many researchers. Let *R* be a *n*-graded ring and *M* a *n*-graded *R*-module. Then, Stanley conjectured that depthð Þ *M* ≤sdepthð Þ *M :* He also conjectured that each Cohen-Macaulay simplicial complex is partition-able. In this chapter, we study the relation between vertex decomposability of some simplicial complexes and Stanley's conjectures.

**Keywords:** vertex decomposable, simplicial complex, Matroid, path

#### **1. Introduction**

Let *R* ¼ *K x*½ � 1, … , *xn* , where *K* is a field. Fix an integer *n*≥ *t*≥2 and let *G* be a directed graph. A sequence *xi*1 , … , *xit* of distinct vertices is called a path of length *t* if there are *t* � 1 distinct directed edges *e*1, … ,*et*�1 where *ej* is a directed edge from *xi j* to *xi j*þ1 . Then, the path ideal of *G* of length *t* is the monomial ideal *It*ð Þ¼ *G xi*1 … *xit* : *xi*1 , … , *xit* ð is a path of length t in G) in the polynomial ring *R* ¼ *K x*½ � 1, … , *xn* . The distance *d x*ð Þ , *y* of two vertices *x* and *y* of a graph *G* is the length of the shortest path from *x* to *y*. The path complex Δ*t*ð Þ *G* is defined by

Δ*t*ð Þ¼ *G xi*1 , … , *xit* f g : *xi*1 , … , *xit* h i is a path of length *t* in *G :*

#### *Advanced Topics of Topology*

Path ideals of graphs were first introduced by Conca and De Negri [1, 2] in the context of monomial ideals of linear type. Recently, the path ideal of cycles has been extensively studied by several mathematicians. In [3], it is shown that *I*2ð Þ *Cn* is sequentially Cohen-Macaulay, if and only if, *n* ¼ 3 or *n* ¼ 5. Generalizing this result, in [4], it is proved that *It*ð Þ *Cn* , (*t*> 2), is sequentially Cohen-Macaulay, if and only if *n* ¼ *t* or *n* ¼ *t* þ 1 or *n* ¼ 2*t* þ 1. Also, the Betti numbers of the ideal *It*ð Þ *Cn* and *It*ð Þ *Ln* is computed explicitly in [5]. In particular, it has been shown that [6, 7]:

**Theorem 1.1** ([1, Corollary 5.15])**.** *Let n, t, p and d be integers such that n* ≥*t*≥2*, n* ¼ ð Þ *t* þ 1 *p* þ *d, where p*≥ 0 *and* 0≤*d*<ð Þ *t* þ 1 *. Then,*

i. *The projective dimension of the path ideal of a graph cycle Cn or line Ln is given by,*

$$\operatorname{pd}(I\_l(\mathbb{C}\_n)) = \begin{cases} 2p, & d \neq 0 \\ 2p - 1, & d = 0 \end{cases} \qquad \operatorname{pd}(I\_l(L\_n)) = \begin{cases} 2p - 1, & d \neq t \\ 2p, & d = t \end{cases} \tag{1}$$

ii. *The regularity of the path ideal of a graph cycle Cn or line Ln is given by,*

$$\begin{aligned} \text{reg}(I\_l(\mathbf{C}\_n)) &= (t - \mathbf{1})p + d + \mathbf{1} \\\\ \text{reg}(I\_l(L\_n)) &= \begin{cases} p(t - \mathbf{1}) + \mathbf{1}, & d < t \\\\ p(t - \mathbf{1}) + t, & d = t \end{cases} \end{aligned} \tag{2}$$

In [8] it has been shown that Δ*t*ð Þ *G* is a simplicial tree if *G* is a rooted tree and *t*≥ 2. One of interesting problems in combinatorial commutative algebra is the Stanley's conjectures. The Stanley's conjectures are studied by many researchers. Let *R* be a *n*-graded ring and *M* a *n*-graded *R*-module. Then, Stanley [9] conjectured that

$$\text{depth}(M) \le \text{sdepth}(M) \tag{3}$$

He also conjectured in [10] that each Cohen-Macaulay simplicial complex is partitionable. Herzog, Soleyman Jahan, and Yassemi in [11–14] showed that the conjecture about partitionability is a special case of the Stanley's first conjecture. In this chapter, we first study algebraic properties of Δ*t*ð Þ *Cn* . In Section 1, we recall some definitions and results, which will be needed later. In Section 2, for all *t*>2 we show that the following conditions are equivalent:

$$\mathbf{v}. \ n = t \text{ or } t + \mathbf{1}.$$

(see Theorem 2.6).

In Section 3, for all *t* ≥2 we show that Δ*t*ð Þ *G* is vertex decomposable if and only if *G* ¼ *H p*ð Þ , *n*, *q* or *G* ¼ *H p*ð Þ , *n* . In Section 4, vertex decomposability path complexes of Dynkin graphs are shown. In Section 5 as an application of our results, we show

that if *n* ¼ *t* or *t* þ 1 then Δ*t*ð Þ *Cn* is partitionable and Stanley's conjecture holds for *K*½ � Δ*t*ð Þ *Cn* and *K*½ � Δ*t*ð Þ *G* , where *G* ¼ *H p*ð Þ , *n*, *q* or *G* ¼ *H p*ð Þ , *n* .

#### **2. Preliminaries**

In this section, we recall some definitions and results which will be needed later. **Definition 2.1.** *A simplicial complex* Δ over a set of vertices *V* ¼ f g *x*1, … , *xn* , is a collection of subsets of *V*, with the property that:

a. f g *xi* ∈ Δ, for all *i*;

b. if *F* ∈ Δ, then all subsets of *F* are also in Δ (including the empty set).

An element of Δ is called a *face* of Δ and complement of a face *F* is *V*n*F* and it is denoted by *Fc* . Also, the complement of the simplicial complex Δ ¼ h i *F*1, … , *Fr* is Δ*c* ¼ *Fc* 1, … , *Fc r* � �. The *dimension* of a face *F* of Δ, dim *F*, is j j *F* � 1 where j j *F* is the number of elements of *F* and dim ∅ ¼ �1. The faces of dimensions 0 and 1 are called *vertices* and *edges*, respectively. A *non-face* of Δ is a subset *F* of *V* with *F* ∉ Δ. We denote by N ð Þ Δ , the set of all minimal non-faces of Δ. The maximal faces of Δ under inclusion are called *facets* of Δ. The *dimension* of the simplicial complex Δ, dim Δ, is the maximum of dimensions of its facets. If all facets of Δ have the same dimension, then Δ is called *pure*.

Let Fð Þ¼ Δ *F*1, … , *Fq* � � be the facet set of Δ. It is clear that Fð Þ Δ determines Δ completely and we write Δ ¼ *F*1, … , *Fq* � �. A simplicial complex with only one facet is called a *simplex*. A simplicial complex Γ is called a *subcomplex* of Δ, if Fð Þ Γ ⊂ Fð Þ Δ .

For *v*∈*V*, the subcomplex of Δ obtained by removing all faces *F* ∈ Δ with *v*∈*F* is denoted by Δn*v*.  The link of a vertex *v*∈*V* is simply denoted by linkΔð Þ*v* .

$$\text{link}\_{\Delta}(\boldsymbol{\nu}) = \{ F \in \Delta \, : \, \, \boldsymbol{\nu} \notin F, \quad F \cup \{ \boldsymbol{\nu} \} \in \Delta \}. \tag{5}$$

Let Δ be a simplicial complex over *n* vertices f g *x*1, … , *xn* . For *F* ⊂ f g *x*1, … , *xn* , we set:

$$\mathfrak{x}\_F = \prod\_{\mathfrak{x}\_i \in F} \mathfrak{x}\_i. \tag{6}$$

We define the *facet ideal* of Δ, denoted by *I*ð Þ Δ , to be the ideal of *S* generated by f g *xF* : *F* ∈ Fð Þ Δ . The *non-face ideal* or the *Stanley-Reisner ideal* of Δ, denoted by *I*Δ, is the ideal of *S* generated by square-free monomials f g *xF* : *F* ∈ N ð Þ Δ . Also, we call *K*½ � Δ ≔ *S=I*Δ the *Stanley-Reisner ring* of Δ.

**Definition 2.2.** A simplicial complex Δ on f g *x*1, … , *xn* is said to be a matroid if, for any two facets *F* and *G* of Δ and any *xi* ∈ *F*, there exists a *x j* ∈ *G* such that ð Þ *F*nf g *xi* ∪ *x j* � � is a facet of Δ.

**Definition 2.3.** A simplicial complex Δ is recursively defined to be *vertex decomposable*, if it is either a simplex, or else has some vertex *v* so that,

a. Both Δn*v* and linkΔð Þ*v* are vertex decomposable, and

b. No face of linkΔð Þ*v* is a facet of Δn*v*.

A vertex *v* which satisfies in condition (b) is called a *shedding vertex*.

**Definition 2.4.** A simplicial complex Δ is *shellable*, if the facets of Δ can be ordered *F*1, … , *Fs* such that, for all 1≤*i* <*j*≤*s*, there exists some *v*∈*F j*n*Fi* and some *l* ∈f g 1, … , *j* � 1 with *F j*n*Fl* ¼ f g*v* .

A simplicial complex Δ is called disconnected, if the vertex set *V* of Δ is a disjoint union *V* ¼ *V*1 ∪*V*2 such that no face of Δ has vertices in both *V*1 and *V*2. Otherwise, Δ is connected. It is well known that

Matroid ) vertex decomposable ) shellable ) Cohen‐Macaulay

**Definition 2.5.** Given a simplicial complex Δ on *V*, we define Δ∨, the *Alexander dual* of Δ, by

$$
\Delta^\vee = \{ V \backslash F : \quad F \notin \Delta \}.\tag{7}
$$

It is known that for the complex Δ one has *I*Δ∨ ¼ *I* Δ*c* ð Þ. Let *I* 6¼ 0 be a homogeneous ideal of *S* and be the set of non-negative integers. For every *i*∈ ∪ f g0 , one defines:

$$t\_i^S(I) = \max\left\{ j : \quad \rho\_{i,j}^S(I) \neq \mathbf{0} \right\} \tag{8}$$

where *βS i*,*j* ð Þ*I* is the *i*, *j*-th graded Betti number of *I* as an *S*-module. The *Castelnuovo-Mumford regularity* of *I* is given by

$$\text{reg}(I) = \sup \{ t\_i^S(I) - i \, : \, \, i \in Z \}. \tag{9}$$

We say that the ideal *I* has a *d-linear resolution*, if *I* is generated by homogeneous polynomials of degree *d* and *βS i*,*j* ðÞ¼ *I* 0, for all *j* 6¼ *i* þ *d* and *i* ≥0. For an ideal that has a *d*-linear resolution, the Castelnuovo-Mumford regularity would be *d*. If *I* is a graded ideal of *S*, then we write ð Þ *Id* for the ideal generated by all homogeneous polynomials of degree *d* belonging to *I*.

**Definition 2.6.** A graded ideal *I* is componentwise linear if ð Þ *Id* has a linear resolution for all *d*.

Also, we write *I*½ � *d* for the ideal generated by the squarefree monomials of degree *d* belonging to *I*.

**Definition 2.7.** A graded *S*-module *M* is called *sequentially Cohen-Macaulay* (over *K*), if there exists a finite filtration of graded *S*-modules,

$$0 = M\_0 \subset M\_1 \subset \cdots \subset M\_r = M \tag{10}$$

such that each *Mi=Mi*�1 is Cohen-Macaulay, and the Krull dimensions of the quotients are increasing:

$$\dim(M\_1/M\_0) < \dim(M\_2/M\_1) < \dots < \dim(M\_r/M\_{r-1}).\tag{11}$$

The Alexander dual allows us to make a bridge between (sequentially) Cohen-Macaulay ideals and (componetwise) linear ideals.

*Vertex Decomposability of Path Complexes and Stanley's Conjectures DOI: http://dx.doi.org/10.5772/intechopen.101083*

**Definition 2.8** (Alexander duality)**.** For a square-free monomial ideal *I* ¼ *M*1, … , *Mq* ⊂*S* ¼ *K x*½ � 1, … , *xn* , the *Alexander dual* of *I*, denoted by *I* ∨, is defined to be:

$$I^\vee = P\_{M\_1} \cap \dots \cap P\_{M\_q} \tag{12}$$

where, *PMi* is prime ideal generated by *x j* : *x j*j*Mi* .

**Theorem 2.1** ([7, Proposition 8.2.20], [5, Theorem 3])**.** *Let I be a square-free monomial ideal in S* ¼ *K x*½ � 1, … , *xn .*

**Remark 2.1.** Two special cases, we will be considering in this paper, are when *G* is a cycle *Cn*, or a line graph *Ln* on vertices f g *x*1, … , *xn* with edges

$$\begin{aligned} E(\mathbf{C}\_n) &= \{ \{ \mathbf{x}\_1, \mathbf{x}\_2 \}, \{ \mathbf{x}\_2, \mathbf{x}\_3 \}, \dots, \{ \mathbf{x}\_{n-1}, \mathbf{x}\_n \}, \{ \mathbf{x}\_n, \mathbf{x}\_1 \} \}; \\ E(L\_n) &= \{ \{ \mathbf{x}\_1, \mathbf{x}\_2 \}, \{ \mathbf{x}\_2, \mathbf{x}\_3 \}, \dots, \{ \mathbf{x}\_{n-1}, \mathbf{x}\_n \} \} \end{aligned}$$

**Remark 2.2.** All Cohen-Macaulay simplicial complexes of positive dimension are connected.

#### **3. Vertex decomposability path complexes of cycles**

As the main result of this section, it is shown that Δ*t*ð Þ *Cn* is matroid, vertex decomposable, shellable, and Cohen-Macaualay if and only if *n* ¼ *t* or *n* ¼ *t* þ 1. For the proof, we shall need the following lemmas and propositions.

**Lemma 3.1.** *Let* Δ*t*ð Þ *Pn be a simplicial complex on the path Pn* ¼ f g *x*1, … , *xn and* 2≤ *t*≤*n. Then* Δ*t*ð Þ *Pn is vertex decomposable.*

*Proof.* If *t* ¼ *n*, then Δ*n*ð Þ *Pn* is a simplex which is vertex decomposable. Let 2≤ *t*<*n* then one has

$$\Delta\_{\mathbf{t}}(\mathbf{P}\mathbf{n}) = \langle \{\mathbf{x}\_1, \dots, \mathbf{x}\_{\mathbf{t}}\}, \{\mathbf{x}\_2, \dots, \mathbf{x}\_{\mathbf{t}+1}\}, \dots, \{\mathbf{x}\_{\mathbf{n}-\mathbf{t}+1}, \dots, \mathbf{x}\_{\mathbf{n}}\} \rangle. \tag{13}$$

So Δ*t*ð Þn *Pn xn* ¼ f g *x*1, … , *xt* , f g *x*2, … , *xt*þ1 , … , *xn*�*t* h i f g , … , *xn*�1 . Now, we use induction on the number of vertices of *Pn* and by induction hypothesis Δ*t*ð Þn *Pn xn* is vertex decomposable. On the other hand, it is clear that *link*Δ*t*ð Þ *Pn* f g *xn* ¼ h i f g *xn*�*t*þ1, … , *xn*�1 . Thus, *link*Δ*t*ð Þ *Pn* f g *xn* is a simplex which is not a facet of Δ*t*ð Þn *Pn xn*. Therefore, Δ*t*ð Þ *Pn* is vertex decomposable.

**Lemma 3.2.** *Let* Δ2ð Þ *Cn be a simplicial complex on x*f g 1, … , *xn . Then* Δ2ð Þ *Cn is vertex decomposable.*

*Proof.*   Therefore, Δ2ð Þ *Cn* is vertex

decomposable. □ **Lemma 3.3.** *Let* Δ*t*ð Þ *Cn be a simplicial complex on x*f g 1, … , *xn and* 3 ≤*t*≤*n* � 2*. Then* Δ*t*ð Þ *Cn is not Cohen-Macaulay.*

*Proof.* It suffices to show that *I*Δ*t*ð Þ *Cn* ∨ has not a linear resolution. Since *I*Δ*t*ð Þ *Cn* ∨ ¼ *I* Δ*t*ð Þ *Cn c* ð Þ, then one can easily check that *I*Δ*t*ð Þ *Cn* ∨ ¼ *In*�*t*ð Þ *Cn* . By Theorem 1.1 we have

$$\text{reg}\left(I\_{\Delta\_{\mathbb{C}}(\mathbb{C}\_{\mathbb{R}})^{\vee}}\right) = (n - t - \mathbf{1})p + d + \mathbf{1}.\tag{15}$$

Since 3 ≤*t*≤*n* � 2 then one has *reg I*Δ*t*ð Þ *Cn* ∨ 6¼ *n* � *t* and by Theorem 2.1 Δ*t*ð Þ *Cn* is not Cohen-Macaulay.

**Proposition 3.1.** *Let* Δ*t*ð Þ *Cn be a simplicial complex on x*f g 1, … , *xn and t*≥ 3*. Then* Δ*t*ð Þ *Cn is vertex decomposable if and only if n* ¼ *t or t* þ 1*.*

*Proof.* By Lemma 3.3, it suffices to show that if *n* ¼ *t* or *t* þ 1, then Δ*t*ð Þ *Cn* is vertex decomposable. If *n* ¼ *t*, then Δ*n*ð Þ *Cn* is a simplex which is vertex decomposable.

If *t* ¼ *n* � 1, then we have

$$\Delta\_{n-1}(\mathbb{C}\_n) = \langle \{\mathbf{x}\_1, \dots, \mathbf{x}\_{n-1}\}, \{\mathbf{x}\_2, \dots, \mathbf{x}\_n\}, \{\mathbf{x}\_3, \dots, \mathbf{x}\_n, \mathbf{x}\_1\}, \dots, \{\mathbf{x}\_n, \mathbf{x}\_1, \dots, \mathbf{x}\_{n-2}\} \rangle.$$

Now, we use induction on the number of vertices of *Cn* and show that Δ*n*�1ð Þ *Cn* is vertex decomposable. It is clear that Δ*n*�1ð Þn *Cn xn* ¼ h i f g *x*1, … , *xn*�1 is a simplex, which is vertex decomposable.

On the other hand,

$$\text{link}\_{\Delta\_{n-1}(\mathbb{C}\_{\pi})} \{ \mathbf{x}\_{\pi} \} = \langle \{ \mathbf{x}\_1, \dots, \mathbf{x}\_{n-2} \}, \dots, \{ \mathbf{x}\_{n-1}, \mathbf{x}\_1, \dots, \mathbf{x}\_{n-3} \} \rangle = \Delta\_{n-2} (\mathbf{C}\_{n-1}).\tag{16}$$

By induction hypothesis linkΔ*n*�1ð Þ *Cn* f g *xn* is vertex decomposable. It is easy to see that no face of linkΔ*n*�1ð Þ *Cn* f g *xn* is a facet of Δ*n*�1ð Þn *Cn xn*. Therefore, Δ*n*�1ð Þ *Cn* is vertex decomposable.

**Proposition 3.2.** Δ2ð Þ *Cn is a matroid if and only if n* ¼ 3 *or* 4*.*

*Proof.* If *n* ¼ 3 or 4, then it is easy to see that Δ2ð Þ *Cn* is a matroid. Now, we prove the converse. It suffices to show that Δ2ð Þ *Cn* is not a matroid for all *n* ≥5. We consider two facets f g *x*1, *x*2 and f g *xn*�1, *xn* .  Since f g *x*2, *xn*�1 and f g *x*2, *xn* are not the facets of Δ2ð Þ *Cn* . So Δ2ð Þ *Cn* is not matroid for all *n*≥5.

For the simplicial complexes, one has the following implication:

#### **Matroid** ) **vertex decomposable** ) **shellable** ) **Cohen** � **Macaulay**

Note that these implications are strict, but by the following theorem, for path complexes, the reverse implications are also valid.

**Theorem 3.1.** *Let t*≥3*. Then the following conditions are equivalent*:

i. Δ*t*ð Þ *Cn is matroid;*

ii. Δ*t*ð Þ *Cn is vertex decomposable;*

v. *n* ¼ *t or t* þ 1.

*Vertex Decomposability of Path Complexes and Stanley's Conjectures DOI: http://dx.doi.org/10.5772/intechopen.101083*

*Proof.* (i)) (ii), (ii)) (iii) and (iii)) (iv) is well known. (iv)) (v): Follows from Lemma 3.3 and Proposition 2.1. (v)) (i): If *n* ¼ *t*, then Δ*t*ð Þ *Cn* is a simplex which is a matroid. If *n* ¼ *t* þ 1, then

$$\Delta\_l(\mathcal{C}\_n) = \langle \{\mathbf{x}\_1, \dots, \mathbf{x}\_l\}, \{\mathbf{x}\_2, \dots, \mathbf{x}\_{l+1}\}, \{\mathbf{x}\_3, \dots, \mathbf{x}\_{l+1}, \mathbf{x}\_1\}, \dots, \{\mathbf{x}\_{l+1}, \mathbf{x}\_1, \dots, \mathbf{x}\_{l-1}\} \rangle.$$

For any two facets *F* and *G* of Δ*t*ð Þ *Cn* one has ∣*F* ∩ *G*∣ ¼ *t* � 1. We claim that for any two facets *F* and *G* of Δ*t*ð Þ *Cn* and any *xi* ∈ *F*, there exists a *x j* ∈ *G* such that ð Þ *F*nf g *xi* ∪ *x j* � � is a facet of Δ*t*ð Þ *Cn* . We have to consider two cases. If *xi* ∈*F* and *xi* ∉ *G*, then we choose *x j* ∈ *G* such that *x j* ∉ *F*. Thus, ð Þ *F*nf g *xi* ∪ *x j* � � ¼ *G* which is a facet of Δ*t*ð Þ *Cn* .

For other case, if *xi* ∈ *F* and *xi* ∈ *G*, then we choose *x j* ∈ *G* such that *x j* is the same *xi*. Therefore, ð Þ *F*nf g *xi* ∪ f g *xi* ¼ *F* is a facet of Δ*t*ð Þ *Cn* , which completes the proof.

#### **4. Vertex decomposability path complexes of trees**

As the main result of this section, for all *t*≥2, we characterize all such trees whose Δ*t*ð Þ *G* is vertex decomposable. Let *H p*ð Þ , *n*, *q* denote the double starlike tree obtained by attaching *p* pendant vertices to one pendant vertex of the path *Pn* and *q* pendant vertices to the other pendant vertex of *Pn*. Also, let *H p*ð Þ , *n* be graph obtained by attaching *p* pendant vertices to one pendant vertex of the path *Pn*.

**Remark 4.1.**  Then, Δ*t*ð Þ *H*ð Þ 2, *n* is vertex decomposable for all *t*≥ 2.

*Proof.* By Lemma 3.1 proof is trivial.

**Proposition 4.1.** *Let Pn* ¼ f g *x*1, … , *xn be a path on vertices x*f g 1, … , *xn and H p*ð Þ , *n be a graph obtained by attaching p pendant vertices to pendant vertex xn. Then* Δ*t*ð Þ *H p*ð Þ , *n is vertex decomposable for all t*≥2*.*

*Proof.* We prove the proposition by induction on *p* the number of pendant vertices to pendant vertex *xn* of *Pn*. If *p* ¼ 0 or 1, then *H p*ð Þ , *n* is a path and by Lemma 2.1 Δ*t*ð Þ *H p*ð Þ , *n* is vertex decomposable. If *p* ¼ 2, then by remark 4.1 Δ*t*ð Þ *H p*ð Þ , *n* is vertex decomposable. Now, let *p*> 2 and *y*1, … , *yp* n o be *p* pendant vertices to pendant vertex *xn* of *Pn*, then one has

$$H(p, n) \backslash \{y\_1\} = H(p-1, n) \tag{17}$$

and

$$
\Delta\_l(H(p,n)) \backslash \{\mathcal{y}\_1\} = \Delta\_l(H(p-1,n)).\tag{18}
$$

Therefore by induction hypothesis Δ*t*ð Þ *H p*ð Þ � 1, *n* is vertex decomposable. So Δ*t*ð Þn *H p*ð Þ , *n y*1 � � is vertex decomposable. If *t* ¼ 3, then we have

$$\text{link}\_{\Delta \chi(H(p,n))} \{ \mathbf{y}\_1 \} = \left\langle \{ \mathbf{x}\_{n-1}, \mathbf{x}\_n \}, \{ \mathbf{y}\_2, \mathbf{x}\_n \}, \dots, \left\{ \mathbf{y}\_p, \mathbf{x}\_n \right\} \right\rangle \tag{19}$$

It is easy to see that linkΔ3ð Þ *H p*ð Þ ,*n y*1 � � is vertex decomposable and *y*1 is a shedding vertex. If *t* ¼ 2 or *t*>3, one has

$$\text{link}\_{\Delta\_{\ell}(H(p,n))} \{ \mathbf{y}\_1 \} = \langle \{ \mathbf{x}\_{n-t+2}, \dots, \mathbf{x}\_n \} \rangle. \tag{20}$$

thus linkΔ*t*ð Þ *H p*ð Þ ,*n y*1 � � is a simplex, which is not a facet of Δ*t*ð Þn *H p*ð Þ , *n y*1 � �, therefore Δ*t*ð Þ *H p*ð Þ , *n* is vertex decomposable.

**Lemma 4.1.** *Let p* ¼ 2 *and q*≥ 2*,Then* Δ*t*ð Þ *H*ð Þ 2, *n*, *q is vertex decomposable for all* 2≤ *t*≤*n* þ 2*.*

*Proof.* Let *H*ð Þ 2, *n*, *q* denote the double starlike tree obtained by attaching two pendant vertices *y*1, *y*2 � � to pendant vertex *x*1 of path *Pn* and *y*0 1, … , *y*0 *q* n o be pendant vertices to pendant vertex *xn* of *Pn*. So by proposition 3.2

Δ*t*ð Þn *H*ð Þ 2, *n*, *q y*1 � � is vertex decomposable. Now, we prove that linkΔ*t*ð Þ *H*ð Þ 2,*n*,*q y*1 � � is vertex decomposable. If *t* ¼ 3, then linkΔ3ð Þ *H*ð Þ 2,*n*,*q y*1 � � ¼ f g *x*1, *x*2 , *x*1, *y*2 � � � � which is vertex decomposable. If *t* ¼ *n* þ 2, then

$$\text{link}\_{\Delta\_{n+2}(H(2,n,q))} \{ y\_1 \} = \left\langle \left\{ \mathbf{x}\_1, \dots, \mathbf{x}\_n, y\_1' \right\}, \left\{ \mathbf{x}\_1, \dots, \mathbf{x}\_n, y\_2' \right\}, \dots, \left\{ \mathbf{x}\_1, \dots, \mathbf{x}\_n, y\_q' \right\} \right\rangle.$$

It is easy to see that linkΔ*n*þ2ð Þ *H*ð Þ 2,*n*,*q y*1 � � is vertex decomposable. If *t* ¼ 2 or 4≤*t* ≤*n* þ 1, then we have linkΔ*t*ð Þ *H*ð Þ 2,*n*,*q y*1 � � ¼ h i f g *x*1, … , *xt*�1 . Thus, linkΔ*t*ð Þ *H*ð Þ 2,*n*,*q y*1 � � is a simplex, which is vertex decomposable. It is clear that *y*1 is a shedding vertex. □

**Proposition 4.2.** *Let Q*1, *Q*2 *be two paths of maximum length k in tree G and y be a leaf of G such that y*∈ *Q*1 ∩ *Q*2*,* ∣*Q*1 ∩ *Q*2∣ ¼ *L. Then* Δ*t*ð Þ *G is not vertex decomposable.*

*Proof.* Suppose *Q*1 ¼ *y*1, *y*2, … , *yk*�*L*, *x*1, *x*2, … , *xL*�1, *y* � � and.

*Q*2 ¼ *y*0 1, *y*0 2, … , *y*0 *k*�*L*, *x*1, *x*2, … , *xL*�1, *y* � � be two paths of length *k* in *G* such that *Q*1 ∩ *Q*2 ¼ f g *x*1, *x*2, … , *xL*�1, *y* and *deg y*ð Þ¼ 1. Since linkΔ*k*ð Þ *G* f g *x*1, … , *xL*�1, *y* is disconnected and pure of positive dimension. By remark 1.11 Δ*k*ð Þ *G* is not Cohen-Macaulay and hence, Δ*k*ð Þ *G* is not vertex decomposable. □

**Proposition 4.3.** *Let G be a double starlike tree such that is not a path. Then* Δ*t*ð Þ *G is vertex decomposable for all* 2≤ *t*≤*n* þ 2*.*

*Proof.* Let *G* ¼ *H p*ð Þ , *n*, *q* denote the double starlike tree obtained by attaching *p* pendant vertices to one pendant vertex of the path *Pn* and *q* pendant vertices to the other pendant vertex of *Pn*. We prove the theorem by induction on *p* the number of pendant vertices to pendant vertex *x*1 of *Pn*. If *p* ¼ 0 or *p* ¼ 1, then by proposition 3.2 Δ*t*ð Þ *G* is vertex decomposable. If *p* ¼ 2, then by Lemma 4.3 Δ*t*ð Þ *G* is vertex decomposable. Now, let *p* >2 and *y*1, … , *yp* n o be *p* pendant vertices to pendant vertex *x*1 of *Pn*. Since *G*n *y*1 � � is again double starlike tree on *p* � 1 pendant vertices. Therefore, by induction hypothesis, Δ*t G*n *y*1 � � � � is vertex decomposable. So Δ*t G*n *y*1 � � � � ¼ Δ*t*ð Þn *G y*1 � � is vertex decomposable. Let *t* ¼ 2, then linkΔ2ð Þ *G y*1 � � ¼ h i f g *x*1 is simplex and vertex decomposable. Let *t* ¼ 3, then linkΔ3ð Þ *G y*1 � � ¼ f g *x*2, *x*1 , *y*2, *x*1 � �, … , *yp*, *x*1 D E n o is vertex decomposable. Let 3< *t*≤*n* þ 1, then linkΔ*t*ð Þ *G y*1 � � ¼ h i f g *x*1, *x*2, … , *xt*�1 is simplex and vertex decomposable. Let *t* ¼ *n* þ 2, then linkΔ*t*ð Þ *G y*1 � � ¼ h *x*1, … , *xn*, *y*1 � �, *x*1, … , *xn*, *y*2 � �, … , f*x*1, … , *xn*, *yp*i is a

path complex of a starlike tree which is vertex decomposable. It is easy to see that no face of linkΔ*t*ð Þ *G y*1 � � is a facet of Δ*t*ð Þn *G y*1 � �. So Δ*t*ð Þ *G* is vertex decomposable. Now, we are ready that prove the main result of this section.

**Theorem 4.1.** *Let G be a tree such that is not a path. Then* Δ*t*ð Þ *G is vertex decomposable for all t*≥2 *if and only if G* ¼ *H p*ð Þ , *n*, *q or H p*ð Þ , *n .*

*Proof.* ð Þ ) We prove by contradiction. Suppose *G* 6¼ *H p*ð Þ , *n*, *q* and *G* 6¼ *H p*ð Þ , *n* . So there exists two paths of maximum length *k* in *G* which contain *L* common

vertices such that one of these vertices is a leaf. Therefore, by proposition 4.2 Δ*k*ð Þ *G* is not vertex decomposable, which is a contradiction. ð Þ ( By proposition 4.1 and Proposition 4.3, the proof is completed.

### **5. Vertex decomposability path complexes of dynkin graphs**

Dynkin diagrams first appeared [15] in the connection with classification of simple Lie groups. Among Dynkin diagrams a special role is played by the simply laced Dynkin diagrams *An*, *Dn*, *E*6, *E*7, and *E*8. Dynkin diagrams are closely related to coxter graphs that appeared in geometry (see [16]).

After that, Dynkin diagrams appeared in many branches of mathematics and beyond, in particular in representation theory. In [17], Gabriel introduced a notion of a quiver (directed graph) and its representations. He proved the famous Gabriel's theorem on representation of quivers over algebraic closed field. Let *Q* be a finite quiver and *Q* the undirected graph obtained from *Q* by deleting the orientation of all arrows.

**Theorem 5.1.** *(Gabriel theorem). A connected quiver Q is of Finite type if and only if the graph Q is one of the following simply laced Dynkin diagrams: An*, *Dn*, *E*6, *E*7 *or E*8*.*

Let *Ln* be a line graph on vertices f g *x*1, … , *xn* and *x j*, *y j* n o be whisker of *Ln* at *x j* with 3≤*j*≤ *n* � 1.

We obtain some condition that Δ*t Ln* ∪*W x j* � � � � is vertex decomposable, as an application of our results, vertex decomposability path complexes of Dynkin graphs are shown. Throughout this section, we assume *Ln* ∪ *W x j* � � be an undirected graph. By Lemma 3.1, we have the following corollary:

**Corollary 5.1.** *Let An be a Dynkin graph and* 2 ≤*t*≤*n. Then* Δ*t*ð Þ *An is vertex decomposable.*

**Proposition 5.1.** *Let Ln be a line graph on vertices x*f g 1, … , *xn and xn*�1, *yn*�1 � � *be whisker of Ln at xn*�1*.*

Then, Δ*t*ð Þ *Ln* ∪*W x*ð Þ *n*�1 is vertex decomposable for all 2≤ *t*≤*n*.

Proof. Then by proposition 4.1 proof is trivial.

**Corollary 5.2.** *Let Dn be a Dynkin graph and n* ≥4*. Then* Δ*t*ð Þ *Dn is vertex decomposable for all* 2≤*t* ≤*n.*

*Proof.* We know that *Dn* ¼ *Ln* ∪*W x*ð Þ *n*�1 . So by proposition 4.3 Δ*t*ð Þ *Dn* is vertex decomposable. □

**Theorem 5.2.** *Let Ln be a line graph on vertices x*f g 1, … , *xn and x j*, *y j* n o *be whisker of Ln at x j with* 3≤*j*≤*n* � 2*.*

Then, Δ*t Ln* ∪*W x j* � � � � is vertex decomposable if and only if 2 ≤*t*≤ 3 or *n* ≥*t*>*α*, where *α* ¼ min *d y j* , *x*1 � �, *d y j* , *xn* n o � � .

*Proof.* We first show that Δ*t Ln* ∪ *W x j* � � � � is not vertex decomposable for all 4≤*t* ≤*α*. It is well known that if Δ is a Cohen-Macaulay simplicial complex, then *link*Δf g*F* is Cohen-Macaulay for each face *F* of Δ.

Also, we know that all Cohen-Macaulay complexes of positive dimension are connected. It is easy to see that *link*Δ*t*ð Þ *Ln* ∪*W x*ð Þ*j x j*, *y j* n o is disconnected and pure of positive dimension.

This implies that Δ*t Ln* ∪*W x j* � � � � is not Cohen- Macaulay and hence Δ*t Ln* ∪ *W x j* � � � � is not vertex decomposable without loss of generality we can assume that *α* ¼ *d y j* , *x*1 � � if *t* ¼ 2 or *n* ≥*t*>*α*, one has:

$$\Delta\_t(L\_\pi \cup W(\mathfrak{x}\_j)) = \langle \{\mathfrak{x}\_1, \dots, \mathfrak{x}\_t\}, \{\mathfrak{x}\_2, \dots, \mathfrak{x}\_{t+1}\}, \dots,$$

$$\{\mathfrak{x}\_{j-1}, \mathfrak{x}\_j, \dots, \mathfrak{x}\_{j+t-2}\}, \{\mathfrak{y}\_j, \mathfrak{x}\_j, \mathfrak{x}\_{j+1}, \dots, \mathfrak{x}\_{j+t-2}\}, \dots, \{\mathfrak{x}\_{n-t+1}, \dots, \mathfrak{x}\_n\})$$

So Δ*t Ln* ∪*W x j* � � � � n*y j* ¼ Δ*t*ð Þ *Ln* and by Lemma [4, 6, 7, 9, 10, 18–20] Δ*t Ln* ∪ *W x j* � � � � n*y j* is vertex decomposable. On the other hand, we have *link*Δ*t*ð Þ *Ln* ∪*W x*ð Þ*j y j* n o ¼ *x j*, *x j*þ1, … , *x j*þ*t*�2 � � � � that is a simplex and vertex decomposable.

If *t* ¼ 3, then

$$\Delta\_3(L\_n \cup W(\mathfrak{x}\_j)) = \langle \{\mathfrak{x}\_1, \mathfrak{x}\_2, \mathfrak{x}\_3\}, \{\mathfrak{x}\_2, \mathfrak{x}\_3, \mathfrak{x}\_4\}, \dots, \mathfrak{x}\_n$$

$$\{\mathfrak{x}\_{j-1}, \mathfrak{x}\_j, \mathfrak{y}\_j\}, \{\mathfrak{y}\_j, \mathfrak{x}\_j, \mathfrak{x}\_{j+1}\}, \dots, \{\mathfrak{x}\_{n-2}, \mathfrak{x}\_{n-1}, \mathfrak{x}\_n\}\rangle$$

and Δ3 *Ln* ∪*W x j* � � � � n*y j* ¼ Δ3ð Þ *Ln* which is vertex decomposable.

It is easy to see that *link*Δ*t*ð Þ *Ln* ∪*W x*ð Þ*j y j* n o ¼ *x j*�1, *x j* � �, *x j*, *x j*þ1 � � � � is vertex decomposable and *y j* is a shedding vertex.

**Corollary 5.3.** *Let E*6 *be a Dynkin graph. Then* Δ*t*ð Þ *E*6 *is vertex decomposable if and only if* 2≤*t* ≤3 *or t* ¼ 5*.*

*Proof.* Since *E*6 ¼ *L*5 ∪ *W x*ð Þ3 , so by Theorem 5.2 the proof is completed.

**Corollary 5.4.** *Let E*7 *be a Dynkin graph. Then* Δ*t*ð Þ *E*7 *is vertex decomposable if and only if* 2≤*t* ≤3 *or* 5≤ *t*≤6*.*

*Proof.* We know that *E*7 ¼ *L*6 ∪*W x*ð Þ3 and the proof follow from Theorem 5.2. **Corollary 5.5.** *Let E*8 *be a Dynkin graph. Then* Δ*t*ð Þ *E*8 *is vertex decomposable if and only if* 2≤*t* ≤3 *or* 5≤ *t*≤7.

#### **6. Stanley decompositions**

Let *R* be any standard graded *K*-algebra over an infinite field *K*, *i:e*, *R* is a finitely generated graded algebra *R* ¼ ⊕*i*≥0*Ri* such that *R*0 ¼ *K* and *R* is generated by *R*1. There are several characterizations of the depth of such an algebra. We use the one that depthð Þ *R* is the maximal length of a regular *R*-sequence consisting of linear forms. Let *xF* ¼ ⊓*i* ∈*Fxi* be a squarefree monomial for some *F* ⊆ ½ � *n* and *Z* ⊆f g *x*1, … , *xn* . The *K*-subspace *xFK Z*½ � of *S* ¼ *K x*½ � 1, … , *xn* is the subspace generated by monomials *xFu*, where *u* is a monomial in the polynomial ring *K Z*½ �. It is called a squarefree Stanley space if *xi* f g : *i*∈ *F* ⊆*Z*. The dimension of this Stanley space is ∣*Z*∣. Let Δ be a simplicial complex on f g *x*1, … , *xn* . A squarefree Stanley decomposition D of *K*½ � Δ is a finite direct sum ⊕*iuiK Zi* ½ � of squarefree Stanley spaces, which is isomorphic as a *n*-graded *K*-vector space to *K*½ � Δ , *i:e*.

$$K[\Delta] \cong \bigoplus\_{i} \mu\_{i} K[Z\_{i}] \tag{21}$$

We denote by sdepthð Þ D the minimal dimension of a Stanley space in D and we define sdepthð Þ¼ *K*½ � Δ max sdepth f g ð Þ D , where D is a Stanley decomposition of *K*½ � Δ . Stanley conjectured in [9] the upper bound for the depth of *K*½ � Δ as the following:

$$\text{depth}(K[\Delta]) \le \text{sdepth}(K[\Delta]) \tag{22}$$

*Vertex Decomposability of Path Complexes and Stanley's Conjectures DOI: http://dx.doi.org/10.5772/intechopen.101083*

Also, we recall another conjecture of Stanley. Let Δ be again a simplicial complex on f g *x*1, … , *xn* with facets *G*1, … , *Gt*. The complex Δ is called partitionable if there exists a partition Δ ¼ ∪ *t i*¼1 *Fi*, *Gi* ½ � where *Fi* ⊆ *Gi* are suitable faces of Δ. Here, the interval *Fi*, *Gi* ½ � is the set of faces f g *H* ∈ Δ : *Fi* ⊆ *H* ⊆ *Gi* . In [10, 18] respectively Stanley conjectured each Cohen-Macaulay simplicial complex is partitionable [16–20]. This conjecture is a special case of the previous conjecture. Indeed, Herzog, Soleyman Jahan, and Yassemi [14] proved that for Cohen-Macaulay simplicial complex Δ on f g *x*1, … , *xn* we have that depthð Þ *K*½ � Δ ≤sdepthð Þ *K*½ � Δ if and only if Δ is partitionable. Since each vertex decomposable simplicial complex is shellable and each shellable complex is partitionable [4, 6, 7, 19, 20]. Then as a consequence of our results, we obtain:

**Corollary 6.1.** *if n* ¼ *t or t* þ 1 *then* Δ*t*ð Þ *Cn is partitionable and Stanley's conjecture holds for K*½ � Δ*t*ð Þ *Cn .*

**Corollary 6.2.** *Let G be a tree such that is not a path. if G* ¼ *H p*ð Þ , *n*, *q or G* ¼ *H p*ð Þ , *n then* Δ*t*ð Þ *G is vertex decomposable for all t*≥2 *and Stanley's conjecture holds for K*½ � Δ*t*ð Þ *G .*

#### **Classification**

2010 Mathematics Subject Classification, 13F20; 05E40; 13F55.

#### **Author details**

Seyed Mohammad Ajdani and Francisco Bulnes\*

1 Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran

2 IINAMEI, Investigación Internacional Avanzada en Matemáticas e Ingeniería, Chalco, Mexico

\*Address all correspondence to<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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## Section 5


**Author details**

Nassrin Malik Aubead Department of the Obstetrics and Gynaecology, Medical College of Hammurabi, University of Babylon, Hilla.Babylon, Iraq

\*Address all correspondence to<EMAIL_ADDRESS><EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**19**

*Role of Sex Hormones in Human Body*

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[5] C. Lauritzen. M. Oettel and E. Schillinger History of Estrogen Research, Physiology and

Mechanisms of Action of Estrogens and Antiestrogens, Experimental Pharmacology, 1999, Volume 135\I, 1,

[6] Sarrel PM (1990) Ovarian hormones

[7] Britta M. Jacobsen and Kathryn B. Horwitz. Progesterone Receptors, their Isoforms and Progesterone Regulated Transcription.Mol Cell Endocrinol. 2012

and the circulation. Maturitas

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[8] Rachel A Davey and Mathis Grossmann. Androgen Receptor Structure, Function and Biology: From Bench to Bedside. Clin Biochem Rev

[9] Jaaskelainen J: Molecular biology of androgen insensitivity. Mol Cell Endocrinol, 2012,352(1-2):4-12.

[10] Shiina H, Matsumoto T, Sato T, et al: Premature ovarian failure in androgen

*DOI: http://dx.doi.org/10.5772/intechopen.95778*

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receptor-deficient mice. Proc Natl Acad

folliculogenesis in female mice lacking androgen receptor. Proc Natl Acad Sci USA, 2004. 101(31):11209-11214.

[13] Sugimura Y, Cunha GR, Donjacour AA: Morphological and histological

degeneration and androgen-induced regeneration in the mouse prostate. Biol

[14] Lai KP, Yamashita S, Vitkus S, et al: Suppressed prostate epithelial development with impaired branching morphogenesis in mice lacking stromal fibromuscular androgen receptor. Mol

[15] Huggins C, Hodges CV: Studies on prostatic cancer: I. The effect of castration, of estrogen and of androgen injection on serum phosphatases in metastatic carcinoma of the prostate.

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[16] Zhou Y, Bolton EC, Jones JO: Androgens and androgen receptor signaling in prostate tumorigenesis. J Mol Endocrinol 2015, 54(1):R15–R29.

[17] Eisermann K, Fraizer G: The androgen receptor and VEGF: mechanisms of androgen-regulated angiogenesis in prostate cancer. Cancers

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[11] Hu YC, Wang PH, Yeh S, et al: Subfertility and defective *Role of Sex Hormones in Human Body DOI: http://dx.doi.org/10.5772/intechopen.95778*


**7. New mechanisms exploration**

In the past recent years, new mechanisms of probiotics on lipid metabolism were proposed. A research by Khedara et al showed lower nitric oxide level has been responsible for hyperlipidemia since endogenous nitric oxide can reduce fatty acid oxidation [104]. Some probiotics had ability to induce nitric oxide synthesis through activation of inducible nitric oxide synthase [105] [106]. Thus, modied NO availability by probiotics play an important role in lipid metabolism.

Moreover, Tanida et al demonstrated that *Lactobacillus paracasei* ST11 could increase adipose tissue lipolysis through enhancing the autonomic nerve activity [107]. In liver, probiotics also exhibited lipid-reducing effects [108]. Ma et al demostrated that VSL#3 probiotics could increase hepatic NKT cell numbers to attenuate high fat diet-induced steatosis [109]. Huang et al found that *L. acidophilus* 4356 could downregulate the Niemann-Pick C1-Like 1 (NPC1L1) level in the duodenum and jejunum of high-fat fed rats [110]. Another recent study by Aronsson et al revealed a new mechanism of *Lactobacillus paracasei* F19 to reduce fat storage by up-regulating levels of Angiopoietin-Like 4 Protein (ANGPTL4) in mice [84].

Omics technology provide a new insight into the mechanisms of lipid metabolism influenced by probiotics. Lee et al demostrated that gene ccpA (encodes catabolite control protein A) had function in cholesterol reduction in vivo by comparation of cholesterolreducing strain *L. acidophilus* A4 and the BA9 mutant strain with no lipid-lowering effect [111]. In addition, six main different expressed proteins involved in these two different strains *in vitro* were identified by proteomic analysis including transcription regulator, FMN-binding protein, major facilitator superfamily permease, glycogen phosphorylase, YknV protein, and fructose/tagatose bisphosphate aldolase.

Microarray analysis of probiotic *L. casei* Zhang effect on liver of high fat diet-fed rats revealed that *L. casei* Zhang administration promote the β-oxidation of fatty acid metabolism through up-regulating five genes expression (Acsl1, Hadh, Acaa2, Acads, and gcdH). Moreover, *L. casei* Zhang could strongly activate expression of glucocorticoid receptor (NR3C1 gene) which might be related to protect against high-fat induced low grade inflammation [112].

Recently, small intestinal proteomes in weanling piglets that respond differently to probiotic (*Lactobacillus fermentum* I5007) and antibiotic (Aureomycin) supplementation in terms of lipid metabolism have shown that probiotic enhanced mucosal SAR1B abundance could prevent weanling piglets from fat malabsorption. More importantly, high mucosal abundance of EIF4A and KRT10 in probiotic-treated piglets may contribute to improve overall gut integrity, suggesting a potential reduction of LPS influx [113].


**4. The concept assessing for hazard of defects by risk criteria**

The above analysis shows that pipeline will invariably contain defects at some stage during its life. These defects will require a "fitness-for-purpose" assessment to determine whether or not to repair the pipeline. The full-scale tests of pipelines with defects and limit state functions method are used for such assessment. The limit state function method allows determining the limit size of defect upon reaching which the pipeline will fail. The limit state function L for pipe with defect can be write as:

$$\mathcal{L}\{P, \mathbf{Q}, \sigma\_f, \mathbf{D}, \mathbf{t}, l\} = l\_r(P, \mathbf{Q}, \sigma\_f, D, \mathbf{t}, l) - l\_i = \mathbf{0} \tag{1}$$

where *P* is operation pressure; *Q* is external loads; *σf* is fracture stress; *D* is outside diameter of pipe; *t* is wall thickness of pipe; *lr* is allowable defect size; *li* is defect size in pipeline.

The defects sizes *li* are established during ILI. The allowable defects sizes *lr* are determined by calculation methods by the specified criteria for the strength and durability of structures, taking in to account the operating conditions and the character of the mechanisms of deformation and destruction [6–10]. It should be emphasized that in these methods the sizes of defects *li* and *lr* are assumed to be deterministic values.

In reality, the defects have inevitable random dispersion of sizes. For detected defects, these are caused by the random nature of the defects, as well as by statistical errors and the probabilistic nature of the operational characteristics (sensitivity and detectability) of non-destructive testing methods [16]. The dispersion of the calculate sizes of defects determined by statistical scattering loads, operating conditions and scattering of mechanical properties. A certain contribution to the possible dispersion of defect sizes is made by idealization of the shapes and schemes of defects. Taking this into account, instead of single-valued sizes in the calculations, it is necessary to use the probability densities distribution functions of defect sizes *f*(*li*) and *f*(*lr*).

Using the functions *f*(*li*) and *f*(*lr*) gives reason to believe that there are always nonzero probabilities *P* presence of defects with sizes *li* larger than *lr* (**Figure 1**):

**Figure 1.** *Probabilistic scheme of the defects hazard analysis.*

*Issues on Risk Analysis for Critical Infrastructure Protection*

$$P(l\_i > l\_r) = \bigcap\_{\substack{l \\ 0 \le l\_r}}^{\infty \text{ og}} f(l\_i) f(l\_r) dl\_r dl\_i \tag{2}$$

inter-field pipelines based on risk criteria. The basis of this method are requirements of standards [7, 9]. The risk is defined as the probability *Rf* negative consequences of pipeline accident, the scale of which is determined by the hazard class. The proposed hazard classes (risk matrix) for inter-field subsea pipelines are presented in **Table 1**. Quantitative economic and environmental damage assess-

The suitability of the pipeline for operation is determined by three-level assessment of the allowable size of defects by risk criteria (**Figure 2**). The first, basic level, determines the allowable defect sizes by the strength characteristics of metal for pipelines exposed to the main loads - internal overpressure and hydrostatic external pressure. The second, extended level, determines the allowable defect sizes by the strength characteristics for metal, taking into account the effect on pipelines of additional longitudinal and bending loads. The third, special level, determines the allowable sizes of cracks, crack-like defects and delamination by the characteristics

The calculations use information about: pipe sizes, location of the pipeline on the

The hazard of pipe defects depends on their shape and size. The sizes of defects are determined by their spatial coordinates *l* = {*lx*, *ly*, *lz*} (**Figure 3**). By shape the defects can be classified into volumetric and flat. For the volumetric defects the size *lx* ≥ *ly* ≥ *lz*, for the flat defects the size *lx* ≥ *ly*> > *lz*. The defect hazard calculations

*Dt* p , ~*lz* ¼ *lz*

� � ¼ *σ f* � *σeRF* ~*lx*,~*lx*

Short-term local disturbance of the state of the ecological environment and/or insignificant material losses. Unscheduled pipeline shut-down and repair.

[*Rf*] 10�2 10�3 10�4 10�5

2*t D*

**Low Middle High Very high**

Neglected Uncritical Critical Catastrophic

Short-term damage to the environment and/or signifycant economic damage. Unscheduled pipeline shutdown and repair.

*RF* ~*lx*,~*lx*

*t* . These relative dimensions are used

� � � *P* ¼ 0 (7)

� � ¼ 0 (8)

Large-scale long-term environmental damage and large economic damage. Long shutdown and pipeline repair.

seabed, loads and impacts; the size, location and types of defects; mechanical

properties, industry standard requirements, and pipe specifications.

in this technique taking into account the classification of defects shape. The limit state function L for pipe volumetric defects may be write as:

> L *P*, *D*, *t*, *σ f* ,~*lx*,~*lx* � � ¼ *σ f*

L *P*, *D*, *t*, *σ f* ,~*lx*,~*lx*

ments are not considered here.

*Defects Assessment in Subsea Pipelines by Risk Criteria DOI: http://dx.doi.org/10.5772/intechopen.94851*

of crack resistance of the metal.

usually use relative defect sizes ~*lx* ¼ *lx*ffiffiffiffi

Negligible environmen-tal and econo-mic impact. Pipeline repa-ir can be post-poned until the planned shutdown.

*Hazard classes of fracture for subsea inter-field pipeline.*

for hoop stress

**Hazard classes**

Failure classes

Level of loss

**Table 1.**

**155**

for equivalent stress

Exceeding the sizes *lr* leads to some losses *C* = *F*(*l*) due to the need to carry out repair operations or leakage and structural failure. Moreover, the larger the defect, the more significant losses can be. It should be emphasized that the losses are also random in magnitude, since it depends on the many technical and socio-economic factors.

Joint analysis of the probabilistic nature of defects, their hazard and possible losses leads to the concept of the admissibility of defects according to risk criteria [11, 18]. The essence of this concept is that the criterion condition for the admissibility of defects is represented in the form:

$$P(l\_i > l\_r) \times C(l) \le [R],\tag{3}$$

where [*R*] is the acceptable risk.

Assuming the defect size *li* as a fixed random variable from (3) we can obtain the following condition for the admissibility of a defect:

$$l\_i \le l\_{[R]} \tag{4}$$

where *l*[*R*] is the size defect at which the risk *R* is acceptable.

Due to the unresolved problem of assessment and statistical analysis of losses, currently, sufficiently substantiated proposals for determining the allowable risk have not been developed. As a rule, losses are categorized into some qualitative classes: negligible, acceptable, unacceptable, etc. [19, 20]. Each class of losses is associated with a certain acceptable level of its probabilities [*Rf*]. Taking this into account, instead of (3), one can go to a simpler form of assessing the admissibility of defects by risk criteria, which does not require a direct assessment of damages, namely:

$$P(l\_i > l\_r) = \int\_0^{l\_i} f(l\_r) dl\_r \le \left[ R\_f \right] \tag{5}$$

On this basis, similarly to (5), the following condition for the admissibility of defects can be written:

$$l\_i \le l\_{\left[ \mathcal{R}\_f \right]} \tag{6}$$

where *l*[*Rf*] is the size of the defect at which the probability of losses belongs to a given class.

Expressions (4) and (6), in fact, are a semi-probabilistic solution to problems (3) and (5), since they relate fixed random variables, one of which has a given probabilistic support.


**Application of Wavelet Transform and Artificial Neural Network to Extract Power Quality Information from Voltage Oscillographic Signals in Electric Power Systems**

R. N. M. Machado1, U. H. Bezerra2, M. E. L Tostes2, S. C. F. Freire1 and L. A. Meneses1 *1Federal Institute of Technological Education, Belém, Pará 2Federal University of Pará, Belém, Pará*

 *Brazil*

## **1. Introduction**

176 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

Mei, K. Rovnyak, S. M. & Ong, C-M. (2006). Dynamic Event Detection Using Wavelet

Mertins, A. (1999). *Signal analysis: Wavelets, Filter Banks, Time-Frequency, Transforms and Applications,* John Wiley&Sons Ltd, ISBN<PHONE_NUMBER>, New York, USA Novosel, D. Madani, V. Bhargava, B. Khoi, V. & Cole, J. (2007). Dawn of the grid

Omerhodzic, I. Avdakovic, S. Nuhanovic, A. & Dizdarevic K. (2010). Energy Distribution of

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Polikar, R. (1999). The Story of Wavelets. *Proceedings of The* IMACS/IEEE CSCC'99, Athens,

Radunovic, D. (2005). *Talasići*, Akademska misao, ISBN 86-7466-190-4, Beograd, Srbija Teofanov, N. (2001). Wavelets - a sentimental history, manuscript of the lecture given on 22. XI 2001. , Department of mathematics and informatics, Novi Sad, Serbia Vetterli, M. & Kovacevic, J. (1995). *Wavelets and subband coding*, Prentice-Hall, Inc., ISBN 0-

*Biological and Life Sciences*, Vol. 6, No. 4, pp. 210-215, 2010

Springer, ISBN 978-0-387-76535-8, New York, USA

Montreal, Canada, June 18-22, 2006

(December 2007), ISSN 1540-7977

Greece, July, pp. 5481-5486, 1999

13-097080-8, New York, USA

X, New York, USA

Analysis, *Proceedings of IEEE PES General Meeting,* pp. 1-7, ISBN 1-4244-0493-2,

synchronization, *IEEE Power and Energy Magazine*, Vol. 6, No. 1, pp. 49 – 60

EEG Signals: EEG Signal Wavelet-Neural Network Classifier. *International Journal of*

Post-operation contingencies analysis in electrical power systems is of fundamental importance for the system secure operation, and also to maintain the quality of the electrical energy supplied to consumers. The electrical utilities use equipments as Digital Disturbance Registers (DDR), and Intelligent Electronics Devices (IED) for faults monitoring, and diagnosis about the electrical power systems operation and protection. In general, the DDR and IED are intended to monitor the protection system performance and detect failures in equipments and transmission lines, and also generate analog and digital oscillographic registers that better characterize the disturbing events.

The oscillographic signals often analyzed in the post-operation centers are those generated by events that typically cause the opening of transmission lines due to the action of protective relays. So, these records are analyzed in detail to determine the causes and consequences of these occurrences within the electrical system. Although the software used in the post-operation centers presents numerous features for the evaluation of the recorded signals, the selection of the signals to be analyzed is done in a manual way, which leads to an analysis in an individual basis, and many of the oscillographic records that could help analyzing the occurrences are not evaluated due to the long time that would be spent to select them manually.

Another aspect to be noted is that the oscillographic records remain stored in the post-operation centers for time periods ranging from months to years. These records contain signals acquired in different parts of the electrical system, and the vast majority of them are no longer being considered in the analysis. These data, however, may contain important information about the behavior and performance of the electrical system that may precisely characterize the power quality problem due to a failure or disturbance.

Application of Wavelet Transform and Artificial Neural Network to

that have become the cornerstone of wavelet applications today.

**2.1 Wavelet analysis**

Daubechies wavelet, db8.

Fig. 1. The Daubechies wavelet, db8.

**2.2 Discrete wavelet transform**

wavelet.

Extract Power Quality Information from Voltage Oscillographic Signals in Electric Power Systems 179

The mathematics main branch leading to wavelet analysis began with Joseph Fourier (1807) with his frequency analysis theory, known as Fourier analysis. The first wavelet mention appears in the appendix of A. Haar's thesis (1909). Paul Levy a 1930's physicist, investigating the Brownian motion, found that the Haar basis functions are superior to the Fourier basis functions for studying small and complicated details in the Brownian motion. In 1980, Grossman and Morlet, broadly defined wavelets in the context of quantum physics, providing a way of thinking about wavelets based on physical intuition. In 1985, Stephane Mallat gave wavelets an additional advance. Through his work in digital signal processing, he discovered some relationships among quadrature mirror filters - QMF, pyramidal algorithm, and orthogonal wavelet basis. Based partially on these results, Y. Meyer built the first non-trivial wavelets, which unlike the Haar wavelet, the Meyer wavelets are continuously differentiable, but do not have compact support. Years later, Ingrid Daubechies used Mallat's work to build a set of wavelets with orthogonal basis functions

The wavelet transform is a technique similar to the windowed Fourier transform with the difference that the window width is variable. The wavelet analysis allows the use of large time intervals when it is desired to get low frequency information and shorter time intervals when the interest is to obtain high frequency information. Unlike Fourier analysis that uses sines and cosines, wavelet analysis uses wavelets. Figure 1 shows as an example, the

Wavelets sets are employed to approximate signals, and each set consists of scaled versions (compressed or expanded) and translated (time shifted) from a single wavelet, called mother

In the discrete wavelet transform the term "discrete" applies only to the parameters in the transformed domain, that is, scales and translations, and not to the independent variable time, of the function being transformed. The discrete wavelet transform provides a set of coefficients corresponding to points on a grid or two-dimensional lattice of discrete points in the time-scale domain. This grid is indexed by two integers, the first, denoted by *m* ,

One of the main difficulties in using measurements, obtained by DDR, in the evaluation of power quality as compared with those obtained by power quality monitors, is that many of the signal processing stages are not performed automatically by the first. For the oscillographic records to be useful as power quality indicators, it is first necessary to obtain certain parameters to classify the recorded signals according to the event type that has occurred. Considering the case of short duration voltage variations (SDVV), the parameters of interest are the event amplitude and time duration. Obtaining these parameters enables the application of statistical tools as presented in (Bollen, 2000), for results analysis and visualization, which allows having information about the electrical system behavior at certain time intervals, for example, months or years.

Another difficulty, perhaps the most critical, is the large volume of data obtained from oscillographic monitoring. Many of these recorded signals are due to switching maneuvers, or due to spurious signals or noise, without characterizing voltage changes in the electrical system. For this large amount of data to be evaluated, it is necessary that an automatic classification method be used so that only signals with the desired characteristics are used to determine the parameters of interest. This aspect is highlighted in several publications which present new methods for classification and characterization using digital signal processing and computational intelligence tools (Angrisani et al, 1998; Santoso et al, 2000a; Santoso et al, 2000b and Huang et al, 2002; Machado et al, 2009; Rodriguez et al, 2010 ).

The first use of wavelet transform in power systems is credited to (Ribeiro, 1994). In recent years, wavelet transform - WT, a powerful tool for digital signal processing, has been proposed as a new technique for monitoring and analysis of different disturbances types in power systems (Machado et al, 2009; Mokryani, 2010; A. Rodriguez et al, 2010; Gong Jing, 2010, 2011). Wavelets, along with computational intelligence techniques like artificial neural networks and fuzzy logic, have been used successfully in automatic classification of power quality problems. (Machado et al, 2009; Mokryani, 2010; Rodriguez et al, 2010)

The present work aims to develop an automated system for classifying power quality problems with respect to the fault type that has occurred and the electric phase involved, and quantify SDVV in electrical power systems from the available oscillography in the electrical utilities post-operation centers, to form a parameter database characterizing power quality problems. The proposed methodology uses the wavelet transform to obtain a characteristic pattern to represent the phenomenon and a probabilistic neural network for classification.


**3. Human impact on plant genetic resources and options for their maintenance**

There is rising awareness that habitat loss is associated with loss of genetic diversity; however the basic cause-effect relationships underlying the ecological roles of biodiversity are still poorly understood [22]. Major tropical wilderness areas show strong coincidence of biodiver‐ sity, people and concerns over watershed functions and are in a state of decrease or degrada‐ tion as natural forests are converted into other land use classes and little area transforms back to natural forests or is reforested with plantations [23]. Various drivers are important for that. There is a high negative correlation of population density on species richness and tree diversity for tropical rainforests [24] but neither population nor poverty is the sole and major reason for land use and land cover change. It is driven by people's reaction to economic opportunities and constraints, created by local as well as national markets and policies. Additionally, global forces become main determinants as they intensify or weaken local factors [25].

Sustainability in agriculture suggests a focus on both improved understanding of the benefits of ecological and agronomic management, manipulation and redesign, and genotype im‐ provements through the full range of modern biological approaches. A sustainable manage‐ ment of agro-ecosystems which includes aspects such as energy flows, nutrient cycling, human impacts and resilience may enhance reshaping of agriculture in landscapes. Sustainable cropping systems will influence food production, pesticide use and carbon stocks in a positive way. But the development of national and international policies supporting a stronger expansion of such systems in developed as well as developing countries still remains a major challenge [26]. Key drivers in the decline of biodiversity, its conservation and ecosystem services are the increased use of pesticides, herbicides and fertilizers, an increased homoge‐ neity of landscapes associated with regional and farm-level specialization, drainage of water logged fields, loss of marginal and un-cropped habitat patches, and reduced fallow periods. Additionally, the intensification of agriculture has been fundamental to the degradation of ecosystem services and increases both, the production of greenhouse gases and a reduction of carbon sequestration [27].

Looking at the human impact on the earth system, nine planetary boundaries can be defined whereby the transgression of one or more of them is considered harmful due to abrupt environmental changes within continental- to planetary-scale systems. Three of them - climate change, the global nitrogen cycle and the rate of biodiversity loss - have already surpassed their threshold levels leading to consequences for global sustainability [28]. Strong negative impacts upon ecosystems are expected when the increasing annual global mean temperature rises above the pre-industrial mean by 2°C or more, especially in biodiversity hotspots [29].

The supply of non-timber forest products, timber and other services by forests resume a safety net function for rural populations providing both, income and jobs. Hence, fostering the role of forests in the political debate is essential, not only when looking at mitigation and adaption strategies for climate change but also for achieving sustainable developmental goals [30]. Experiences from Central Africa involving a wide range of stakeholders showed that focused research on priority sectors for poverty reduction are likely to raise public awareness of the forests' role and contribution to mitigate and to adapt to climate change at regional and national levels. A key for improving coherence and effectiveness of forest management policies is establishing the link between forests and climate change adaptation [30].

Plant genetic resources are essential for farmers to cope with future challenges, a feature emphasized by the FAO Commission on Genetic Resources for Food and Agriculture. New within-crop diversity will be needed to adapt to future conditions, and even new crops will be required under extreme conditions to reduce risks induced by climate change [4].

Despite progress in the past, crop wild relatives and underused species for food and agriculture need to be secured. The need for adapted germplasm is urgent and requires characterization, evaluation, and the availability of materials but gaps in *ex-situ* collections of selected crop gene pools are huge. The current focus on major crops leads to concerns due to a lack of *in-situ* and on-farm conservation. This neglects the importance of genetic, species, and ecosystems diversity. Currently, six million plant accessions are conserved in gene banks worldwide, representing a very limited number of species. Half of them are improved cultivars or breeders' lines. Only one third, however, represents landraces or old cultivars. About 15% refer to wild relatives of crop species and weeds. Minor crops and underutilized species are largely underrepresented in these collections, particularly primitive cultivars and wild relatives from the centres of origin's diversity and cultivation. Another obstacle is that only a third of all gene bank accessions have been fully characterized [1].

However, for many other crops, especially neglected or underutilized species and wild relatives of crops, comprehensive collections still do not exist and considerable gaps remain to be filled. There is also need for a better communication, collaboration and partnerships among institutions dealing with the management of plant genetic resources, from conservation to plant breeding and seed systems [4].

services are the increased use of pesticides, herbicides and fertilizers, an increased homoge‐ neity of landscapes associated with regional and farm-level specialization, drainage of water logged fields, loss of marginal and un-cropped habitat patches, and reduced fallow periods. Additionally, the intensification of agriculture has been fundamental to the degradation of ecosystem services and increases both, the production of greenhouse gases and a reduction of

Looking at the human impact on the earth system, nine planetary boundaries can be defined whereby the transgression of one or more of them is considered harmful due to abrupt environmental changes within continental- to planetary-scale systems. Three of them - climate change, the global nitrogen cycle and the rate of biodiversity loss - have already surpassed their threshold levels leading to consequences for global sustainability [28]. Strong negative impacts upon ecosystems are expected when the increasing annual global mean temperature rises above the pre-industrial mean by 2°C or more, especially in biodiversity hotspots [29].

The supply of non-timber forest products, timber and other services by forests resume a safety net function for rural populations providing both, income and jobs. Hence, fostering the role of forests in the political debate is essential, not only when looking at mitigation and adaption strategies for climate change but also for achieving sustainable developmental goals [30]. Experiences from Central Africa involving a wide range of stakeholders showed that focused research on priority sectors for poverty reduction are likely to raise public awareness of the forests' role and contribution to mitigate and to adapt to climate change at regional and national levels. A key for improving coherence and effectiveness of forest management policies is

Plant genetic resources are essential for farmers to cope with future challenges, a feature emphasized by the FAO Commission on Genetic Resources for Food and Agriculture. New within-crop diversity will be needed to adapt to future conditions, and even new crops will

Despite progress in the past, crop wild relatives and underused species for food and agriculture need to be secured. The need for adapted germplasm is urgent and requires characterization, evaluation, and the availability of materials but gaps in *ex-situ* collections of selected crop gene pools are huge. The current focus on major crops leads to concerns due to a lack of *in-situ* and on-farm conservation. This neglects the importance of genetic, species, and ecosystems diversity. Currently, six million plant accessions are conserved in gene banks worldwide, representing a very limited number of species. Half of them are improved cultivars or breeders' lines. Only one third, however, represents landraces or old cultivars. About 15% refer to wild relatives of crop species and weeds. Minor crops and underutilized species are largely underrepresented in these collections, particularly primitive cultivars and wild relatives from the centres of origin's diversity and cultivation. Another obstacle is that only a third of all gene

However, for many other crops, especially neglected or underutilized species and wild relatives of crops, comprehensive collections still do not exist and considerable gaps remain to be filled. There is also need for a better communication, collaboration and partnerships

be required under extreme conditions to reduce risks induced by climate change [4].

establishing the link between forests and climate change adaptation [30].

bank accessions have been fully characterized [1].

carbon sequestration [27].

128 Agroecology

Since farmers know best which materials meet their needs and are enthusiastic seekers of new varieties, "participatory plant breeding" represents a promising approach to enhancing agrobiodiversity [3]. Their participation in the whole process from the selection of plant genetic material up to development of cropping systems would improve and help to meet their needs, while also sustaining food security and alleviating poverty. Plant breeding of major crops, however, often lacks this participation, neglecting marginal site conditions and fostering high input demanding improved varieties.

Genetic diversity is essential for improving crops already in use but also for developing potential novel crops. The successful use of plant genetic resources and their sustained production depend to a large degree on access to genetically diverse germplasm. Free international exchange of germplasm will contribute significantly to the worldwide develop‐ ment of new industrial crops [31]. This is another challenge which needs to be solved in an equitable and fair way. Often indigenous knowledge is highly esteemed by people but once it comes to payment for such services people are reluctant to do so. Contracts for compensating creativity of farmers and a framework for rewarding grass root creativity and innovation are possible solutions [32], while consumers' and traders' responsibility is fostering fair trade, supporting local people in their production of food, feed, fibre and fuel [33]. Four types of collaboration are considered vital for using plant genetic resources in a sustainable way: (i) a national regulatory framework for biodiversity prospecting, (ii) the development of infra‐ structure and technology; (iii) formal contractual relationships among biodiversity's sources, intermediaries and final users; and (iv) the moving of research and development into the source country so as to contribute to its gross national product [16].

There are knowledge gaps with regard to the world's ability to match both a bio-based energy production and maintaining food security at the same time [34]. The main challenge is the competition for arable land and limited fresh water resources associated with a fast growing demand for food, feed, fibre, and fuel worldwide. A solution requires higher crop yields and improving resource-use efficiencies, especially that of nitrogen, and water productivity in production systems over the next two or three decades which is only possible with high external inputs. Cropping systems adapted to climate change and improved stress tolerance of crops are key issues in this context. This can be achieved by genetic improvement of crops and establishing sustainable cropping systems in diverse environments. Integrated assess‐ ments of their impact on resources, environments, and people's welfare can help identifying management options, species and varieties well adopted or most appropriate for specific environments. This will largely depend on the added value regarding specific ambitions, i.e. food, feed, fibre or fuel [34].

Energy wealth in Latin America has so far contributed little to overcome poverty and foster development. The sometimes considerable oil and gas earnings are not used to improve general welfare but skimmed off by the elite. A major problem is that both, energy reserves and energy utilization, are unevenly distributed in Latin America [35]. Using bioenergy is an option for an improved participation of rural communities if political leaders set the right frame.

There is lack of proper research, training, and socio-economic information to produce biofuels in a sustainable way. Hence, research in agriculture has to set a focus on improved crop selection based on the local situation and on management options including cultivation, management of pests and diseases, mechanisation, and harvesting. Furthermore, it is neces‐ sary to adapt cropping systems to local soil conditions and use by-products of biofuel crops to increase the efficiency of nutrient use and decrease negative influences on the environment. This is essential for the sustainable production of bio-fuel crops [36].

There are two main approaches to conserve plant genetic resources:

Both approaches can also be applied to conserve agro-biodiversity where many land-races have vanished since the green revolution in the 1960ies. However, effectively conserving wild biodiversity in agricultural landscapes will require increased research, policy coordination, and strategic support to agricultural communities and conservationists [38]. Research needs to address open questions regarding the minimum size and level of area connectivity required to conserve biodiversity *in-situ* at landscape level. *Ex-situ* conservation is biased by human decisions. But where shall we set priorities for sampling? Who pays for the costs of collecting and sustaining such kind of environmental services? Can value be added by exploring options for development of bio-products? Identifying common priorities in shared natural resource systems, however, is a major step in sharing a common responsibility in addressing climate change and associated problems.