In Chapters 4 and 5, we have discussed various methodsfor finding the solutions of first- and second-orderpartial differential equations (PDEs). Now, weundertake the solution of those linear PDEs, inwhich higher order derivatives occur. Thetheoretical framework and methods of solutiondeveloped for first and second-order linearequations are extendable to a very limited class oflinear equations of higher (arbitrary) order.Generally, it is very difficult to solve any linearPDE, wherein the coefficients are functions ofx and y. Henceforth, we start ourjourney with the description of certain kinds oflinear PDEs, in which the various terms aremultiplied by constants only. We devote Sections 6.1and 6.2 to such linear PDEs. In Sections 6.3–6.5, wediscuss some special types of linear equations withvariable coefficients, which are reducible easily toequations with constant coefficients and hencebecome solvable.

6.1 Homogeneous Linear PDEs with ConstantCoefficients

The most general linear PDE with constant coefficientsof order n is of theform

where the coefficients A0,A1,…,An, B0,…, P, Q, Z all are constantsand A0,A1,…,An do not vanishsimultaneously. Recall that an equation of the form(6.1) is called homogeneous whenever f(x,y) = 0. Otherwise, it is callednon-homogeneous.

Conveniently, Eq. (6.1) can be symbolically writtenas

where the polynomial

remains a differential operator with constantcoefficients.

As mentioned in Chapter 2, the mixed partialderivatives of zinvolved in a PDE are assumed to be independent ofthe order of differentiation. Subsequently, for anequation F(D, D')z = f(x, y)with constant coefficients such that F(D,D') = g(D, D') ⋅h(D, D'), we have h(D,D') ⋅ g(D, D') =F(D, D'). This observation canbe described in the form of the following resultthat has an important bearing in the context oflinear PDEs.

Proposition 6.1.Let F(D, D')z = f(x, y)be a linear PDE with constantcoefficients. If the polynomial F(D, D') can be decomposed into some factors, then theorder in which these factors occur isunimportant.

In this section, we discuss the techniques for solvingthe homogeneous linear PDEs with constantcoefficients of arbitrary order. For the sake ofbrevity, we divide this section into foursubsections.

What do you see when you look at Figure 11.1? (And how many children do you see? Many people do not immediately see the infant in the lower right of the image.) What do you imagine the woman is thinking and feeling? How about the children? Do you react emotionally to this image? Often called Migrant Mother, this photograph was taken by the great American photographer Dorothea Lange on a cold and rainy day in February, 1936, in the small town of Nipomo, California (Gordon, 2009). The woman in the photograph had gone to Nipomo to work in the harvest, but unusually cold weather had frozen the crop. That meant no work and no income for her family (Gordon, 2009). When the San Francisco News published this photograph and one other by Lange the following month, readers sent contributions totaling $200,000 (the equivalent in 2020 dollars of $3.7 million) for the stranded farmworkers of Nipomo (Gordon, 2009).

Introduction

The pistil of a flower is exposed to all types of pollen grains in the atmosphere irrespective of whether they belong to the same species or not. However, mere landing of pollen on the stigma is not enough to effect fertilization. As we learnt in the last chapter, there are cellular interactions or cross talk that take place between the pollen and the pistil before successful fertilization. These specific interactions between pollen and pistil facilitate selection of the right type of pollen grains by the pistil and limit fertilization between incompatible gametes.

The inability of a functional male gamete and female gamete to fuse with each other and achieve fertilization is termed as sexual incompatibility. Sexual incompatibility may be interspecific or intraspecific. Following pollination, the ability of a pistil (or stigma) to reject pollen grains from other species is termed as inter-specific incompatibility. This type of incompatibility prevents the formation of inter-specific hybrids and maintains the identity of a biological species. The inter-specific incompatibility is controlled by several genes and is also referred to as heterogenic incompatibility. Interestingly, in nature there are several incidences where pistil carrying functional female gametes are unable to set fruits even when pollinated by viable and fertile self-pollen grains. Scientific investigation have established that the failure of fruit set in these plants is due to genetic factors which impose a physiological barrier to self-fertilization. This phenomenon of failure of a male gamete and a female gamete to achieve self-fertilization is termed as intra-specific incompatibility or more specifically, self-incompatibility (SI). In other words, self-incompatibility is the inability of a fertile hermaphrodite plant to set seeds when self-pollinated. The term self-incompatibility was first coined by Stout (1917); it allows flowering plants to avoid inbreeding and involves genetic mechanisms which prevent self-fertilization and promote out-crossing.

In a self-incompatible plant, whenever its own pollen grains reach stigma either pollen germination or pollen tube growth is terminated which results in failure of seed-set. Yet, there are incidences where self-pollen are able to germinate, and self-pollen tubes are even able to penetrate the ovules. In these cases either fertilization fails to occur, or if at all occurs, the zygote gets aborted after syngamy. This type of SI is called Late Acting Self-incompatibility (LSI).

Introduction

The protective seed habit is a significant feature in the evolutionary success of angiosperms. The seed, encloses an undeveloped miniature plant ‘the embryo’ and acts as a functional unit which links the successive generations. Developmentally, seed is a fertilized ovule and, a typical angiospermous seed consists of an embryo, some storage tissue (mostly endosperm) and a seed-coat. While the embryo and the endosperm are the products of double fertilization, the seed-coat develops from the integument/s of the ovule. Embryos accomplish their early development before seed germination, protected by the surrounding seed coats and sustaining on the stored food in the endosperm. Protection provided to embryo by seed coat increases its chances of survival, and establishment of subsequent generations. Generally, seeds develop as discrete units attached to the inside of the fruit wall through a stalk called the funiculus. However, in many plants, seeds are associated with some other structures that help in their dispersal. In such cases, a single entity of the seed and the structure assisting in dispersal are together described as dispersal units or diaspores. For example, in the members of Asteraceae, the outer integument of the ovule is completely fused with the ovary wall and the diaspore is called a cypsella. Some other examples of diaspores are the seeds with the elaiosomes, achene (dry indehiscent fruits), and caryopsis (fruit type seen in grasses).

Here, one must acknowledge that all structures and processes associated with reproduction in angiosperms are directly responsible for the formation of seed. Seeds perform a wide variety of functions including dispersal, perennation (surviving seasons of stress such as winter), dormancy (a state of arrested development), and most importantly, perpetuation of a plant species.

A huge variation in the size, shape, color, seed coat, weight and dispersal mechanism can be observed among angiospermous seeds. The smallest known seeds are those of orchids which are about 85 micrometers in size and weigh about 0.8 micrograms, thus appearing similar to dust particles. Double coconut or Lodoicea maldivica has the largest (nearly 0.5 meter) and the heaviest (weighing up to 25 kg) known seeds in the world (Fig. 12.1). The size of the seed in a plant depends on the size of the embryo and also on whether the seed at maturity is endospermous or non-endospermous. Seeds in Orchids are small as the endosperm formation is completely suppressed, and also, the embryo is highly reduced.

Introduction

The science of physics generally deals with physical phenomena where one quantity (called an independent variable) is related to another quantity (called a dependent variable) through a mathematical equation. The graphical representation of these data is a convenient tool for deciphering this scientific information. It is of utmost importance that the experimental data should be plotted very carefully so that it is easy to appropriately visualize and interpret the relationship between the dependent and independent variables. For example, it is always advisable to

This chapter introduces the reader to various plotting commands invariably used in this book for developing meaningful graphs. The importance of this chapter lies in the fact that it gives an overview on writing small user-defined functions for generating self-explanatory graphs, instead of writing long codes.

The first method is trivial and is left for the reader to explore. In most of the following chapters, graphs and plots have been formatted using small functions that are executed in a script. The major focus of this chapter is to introduce the reader to this kind of formatting tool. However, for completeness, direct command line instructions have also been mentioned wherever possible.

The layout of this chapter is as follows. The Scilab commands ‘plot’ and ‘plot2d,’ have been used in this book for generating graphs. Section 2.2 starts with highlighting the basic difference between these two commands and manipulating them so that they are on equal footing. This section also focuses on writing small functions for editing the coordinate axes.

In the city or town where you live, are there tent cities such as the one shown in Figure 10.1? Have you or someone you know ever lived in a tent city? How might people be affected by living outside instead of indoors? Why do you think there are tent cities in a nation as wealthy as the United States?

Plants in general and flowering plants (angiosperms) in particular are the essential components for sustenance of life of all non-photosynthetic organisms on our planet. Plants reproduce by asexual as well as sexual means. Asexual reproduction is not congenial for long-term sustenance and evolutionary processes of the species because of genetic uniformity of the progeny. Sexual reproduction which permits genetic recombination is the dominant mode. Although Angiosperms were the last to evolve as land plants, they soon became the most successful and dominant group amongst land plants. Their success is largely due to the mode of their reproduction through the evolution of the flower and the consequent advantages it brought in. For human beings, flowering plants provide most of their essential needs – food, fibres, shelter, medicines, clean air and water. Reproduction is the basis for sustenance of any species. Thus, understanding reproductive biology of flowering plants is important not only from the fundamental point of view but also for their manipulation for human welfare. Reproductive biology of angiosperms is more complex when compared to other groups of plants because of the involvement of the flower. The progress in understanding the structural and functional aspects of reproduction has been very slow.

Initial studies on reproductive biology of angiosperms were largely confined to examining embryological details using fixed and sectioned materials. Enormous data accumulated over the years on the developmental details of the pollen grains, ovules and female gametophyte, double fertilization, embryo and endosperm, seed and fruit development. These advances were taught to the undergraduate and postgraduate students under the title embryology of angiosperms as a part of their curriculum. Following the development of electron microscopy and histochemistry, embryological details were further elaborated by using these techniques. Development of aseptic culture techniques broadened scope for experimental studies on embryological processes leading to a slow but steady understanding of the functional details of embryological structures. These developments were incorporated in some of the books of embryology under a chapter on experimental embryology. However, there was hardly any integrated account of embryological processes in relation to the structure with their function. Pre-fertilization aspects of reproductive biology covering the details of pollen, pistil, and pollen–pistil interactions, which are unique to angiosperms and play a critical role in their successful evolution, were the last to enter the field of embryology of angiosperm.

The inception of interest in understanding mechanisms of plant reproduction is as old as inception of interest in biology. The seminal work and critical observations by Charles Darwin can be regarded as a foundation for establishing a wide interest in pollinators and reproductive biology of angiosperms as a formal subject. In the last few decades, systematic field investigations, advancement of microscopy tools, and molecular techniques have taken the reproductive biology of angiosperms to a new zenith. The scope of the subject is no longer limited to just studying embryo-endosperm development and taxonomic studies but is extended to study the effect of climate change, evolution, conservation of threatened taxa, raising commercial plantations and orchards, pollinator management, seed development, population biology, phyto-geography, and much more. The reproductive biological studies are also closely linked with the understanding of, physiology, genetics and epigenetics of plants.

For a thorough understanding of the subject, a textbook summarizing the basic concepts of plant reproduction integrated with current research, is the need of the hour for both students and instructors. The aim of the present book is to provide a comprehensive account of basic concepts and recent developments in the field of reproductive biology of flowering plants with essential practical exercises. The book extensively covers all the topics from structure of a flower to seed dispersal and presents the concepts with accompanying color photographs and illustrations wherever necessary, to enhance the level of a student's perception. The new, advanced and interesting information is also provided in a box format in each chapter to reinforce learning. An elaborate glossary and questions are provided with each chapter for quick revision and concept enhancement. Boxes summarizing differences between two terms/concepts which students otherwise usually find difficult to comprehend have also been furnished in the book. This book is a blend of theoretical concepts and details of hands-on exercises in the field and laboratory. Methods for field observations, sample observation tables, and suggestions for plant materials to be used for classroom studies/demonstrations pertaining to each concept have also been provided. In addition, the observation sections under practicals are supplemented with the photographs.

Introduction

The fundamental theorem of calculus closely relates integration to differentiation; it implies that integration can be considered as an inverse operation of differentiation, in the sense that, when a continuous function is integrated and then differentiated, it gives back the original function.

Integration refers to the area under the curve defined by a function f in a given interval [a, b]. If x is variable of integration, then the definite integral (y) is given by Eqn. 5.1.

There are innumerable examples in physics that require integral calculus. For instance, simulations of classical mechanics problems may require evaluation of velocity from the acceleration of a body; and displacement of an object from its velocity profile. However, sometimes direct calculations become formidable or intractable, and direct integration rules have to be replaced by approximate numerical methods.

Differentiation refers to finding the rate of change of a dependent quantity (y) with respect to a change in independent quantity (x), i.e. dy/dx . There are countless applications of differential calculus in physics, such as determination of slope and tangent of geometric curves, especially when the rate of change is not constant.

This chapter starts with a discussion on various numerical techniques used for estimating the definite integral of a function. Improper integrals will be discussed in Chapter 6 (on Special Functions). This is followed by a quick overview of the methods of differential calculus which are often used in Scilab.

The layout of this chapter is as follows. In Section 5.2, the built-in Scilab functions dedicated to computation of definite integrals are discussed. User-defined customized Scilab functions based on trapezoidal and Simpson's methods are discussed in Sections 5.3 to 5.5. The method of differentiation is discussed in Section 5.6. The knowledge acquired in all these sections is applied to various advanced physics problems in Section 5.7.

Built-in Scilab Functions for Integration

Scilab has several built-in functions to calculate definite integrals. In this book, two built-in functions have been used for integration, namely,

intg

Consider the definite integral given in Eqn. 5.2.

In order to use the built-in Scilab function, the first step is to define the function that has to be integrated. This is shown in the following.

Introduction

The theory of Fourier series and Fourier integrals is of great importance for a wide range of scientific applications, such as in acoustics, optics and signal processing. The Fourier series is a mathematical representation of a continuous periodic function as an infinite sum of sinusoidal waves. The Fourier analysis is an excellent method to decompose an arbitrary function into sinusoidal components and solve it to get analytical solutions that are otherwise difficult to obtain.

Fourier transform is the frequency domain representation of continuous time signals. It results when the period of a time signal is stretched and allowed to approach infinity.

This chapter introduces the reader to write Scilab programs for computation of Fourier series of various periodic functions. An outline of the chapter is as follows. Section 7.2 discusses the generation of periodic functions. A quick recapitulation of the Fourier series and the significance of harmonics are done in Sections 7.3 and 7.4, respectively. The method of writing Scilab programs for determining Fourier series of various periodic functions has been explained in Section 7.5. In Section 7.6, Fourier transform of some commonly used functions has been discussed. Section 7.7 winds up the chapter with a brief summary of the Fourier analysis. Some practice questions have been given in Section 7.8.

Periodic Functions

A periodic function is a function whose value repeats itself after a regular interval which is called as ‘period’ of that function. As shown in Eqn. 7.1, if function f(x) is periodic and periodicity is d (a non-zero constant number), then for all the values of ‘x’,

Trigonometric functions are common examples of periodic functions. For example, consider the function in Eqns. 7.2. It has a base period equal to π/2.

The following Scilab function generates periodic functions over a given interval. If periodicity of the function is 2T, then, there can be two cases for generating a periodic function.

Case (I) The function is defined within the range [–T, T].

Case (II) The function is defined within the range [0, 2T].

The usefulness of these functions is explained with the help of the following examples.

Example 1: Suppose f(x) is a periodic function in the interval [–2, 2] such that,

The function in Eqn. 7.3 has a periodicity of 4. The following Scilab code generates this periodic function in the interval [–8, 8]. The graph is shown in Figure 7.1.

Introduction

Reproduction is the ultimate goal of every life-form on earth. Accordingly, flowering plants have evolved diverse and versatile strategies to ensure their reproductive success. Broadly, reproduction in higher plants can be divided into two types: sexual and asexual reproduction. Sexual reproduction in vascular plants is complex wherein, multicellular haploid and diploid generations alternate. The diploid sporophyte undergoes meiosis to produce haploid gametes which undergo fusion or syngamy to give rise to seed, the next sporophytic generation. On the other hand, asexual reproduction in plants occurs when a plant produces offspring without meiosis and syngamy. New individuals produced through asexual reproduction are genetically identical to the mother plant. Both sexual and asexual reproduction, have distinct advantages for natural plant populations. Sexual reproduction introduces genetic variability in a population and thus increases the adaptability of species to changing environments. By contrast, asexual reproduction eliminates the cost and the complexity associated with biparental sexual reproduction, and also fixes the genotype of mother plant as offsprings produced are clonal.

When vascular plants reproduce asexually, new individuals may be produced from somatic cells or somatic structures (vegetative reproduction) or through seeds that are produced without fertilization (apomixis or agamospermy). Vegetative reproduction occurs through propagules like bulbils, suckers, and tubers, which are generated from vegetative parts of a plant. Apomixis (away from mixing), is the formation of an embryo and seed from an unreduced gametophyte or sporophyte. Thus, apomixis leads to the formation of a seed without the processes of meiosis (apomeiosis), and fertilization (nuclear fusion). The discovery of apomixis in higher plants is attributed to the observation of a solitary female plant of an Australian species Alchornea ilicifolia (syn. Caelebogyne ilicifolia) by Smith (1841). This female tree would constantly form seeds at the Royal Botanic Gardens in England without any pollen donor around. The term apomixis was introduced by Winkler (1908) to denote “substitution of sexual reproduction by an asexual reproduction process without nuclear and cell fusion”. This led to the use of term apomixis to describe all forms of asexual reproduction in plants (including vegetative reproduction), but this generalization is no longer accepted.

This chapter introduces some general principles of parallel algorithm design. We will consider a few case studies to illustrate broad approaches to parallel algorithms. As already discussed in Chapter 5, the underlying goal for these algorithms is to pose the solution into parcels of relatively independent computation, with occasional interaction. In order to abstract the details of synchronization, we will assume the parallel RAM (PRAM) or the bulk-synchronous parallel (BSP) model to describe and analyze these algorithms. It is a good time for the reminder that going from, say, a PRAM algorithm to one that is efficient on a particular architecture requires refinement and careful design for a particular platform. This is particularly true when “constant time” concurrent read and write operations are assumed. Concurrent reads and writes are particularly inefficient for distributed-memory platforms, and are inefficient for shared-memory platforms as well. It requires synchronization of the processors’ views of the shared memory, which can be expensive.

Recall that PRAM models focus mainly on the computational aspect of algorithm, whereas practical algorithms also require close attention to memory, communication, and synchronization overheads. PRAM algorithms may not always be practical, but they are easier to design than those for more general models. In reality, PRAM algorithms are only the first step toward more practical algorithms, particularly on distributed-memory systems.

Parallel algorithm design often seeks to maximize parallelism and minimize the time complexity. Even if the number of actually available processors is limited, higher parallelism translates to higher scalability in practice. Nonetheless, the work-time scheduling principle (Section 3.5) indicates that low work complexity is paramount for fast execution in practice. In general, if the best sequential complexity of solving the given problem is, say To(n), we would like the parallel work complexity to be O(To(n)). It is a common algorithm design pattern to assume up to To(n) processors and then try to minimize the time complexity. With maximal parallelism, the target time complexity using To(n) processors is O(1). This is not always achievable, and there is often a trade-off between time and work complexity. We then try to reduce the work complexity to O(To(n)), without significantly increasing the time complexity.

Parallel programming is challenging. There are many parts interacting in a complex manner: algorithm-imposed dependency, scheduling on multiple execution units, synchronization, data communication capacity, network topology, memory bandwidth limit, cache performance in the presence of multiple independent threads accessing memory, program scalability, heterogeneity of hardware, and so on. It is useful to understand each of these aspects separately. We discuss general parallel design principles in this chapter. These ideas largely apply to both shared-memory style and message-passing style programming, as well as task-centric programs.

At first cut, there are two approaches to start designing parallel applications:

• 1. Given a problem, design and implement a sequential algorithm, and then turn it into a parallel program based on the type of available parallel architecture.

• 2. Start ab initio. Design a parallel algorithm suitable for the underlying architecture and then implement it.

In either case, performance, correctness, reusability, and maintainability are important goals. We will see that for many problems, starting with a sequential algorithm and then dividing it into independent tasks that can execute in parallel leads to a poor parallel algorithm. Instead, another algorithm that is designed to maximize independent parts, may yield better performance. If a good parallel solution cannot be found – and there do exist inherently sequential problems, for which parallel solutions are not sufficiently faster than sequential ones – it may not be a problem worth solving in parallel.

Once a parallel algorithm is designed, it may yet contain parts that are sequential. Further, the parallel parts can also be executed on a sequential machine in an arbitrary sequence. Such “sequentialization” allows the developer to test parts of a parallel program. If a purely sequential version is already available, or can be implemented with only small effort, it can also serve as a starting point for parallel design. The sequential version can be exploited to develop the parallel application incrementally, gradually replacing sequential parts with their parallel versions. The sequential version also provides performance targets for the parallel version and allows debugging by comparing partial results.