Tuesday, March 10, 2009

Cell Theory

Cell theory refers to the idea that cells are the basic unit of structure in every living thing. Development of this theory during the mid 1600s was made possible by advances in microscopy. This theory is one of the foundations of biology. The theory says that new cells are formed from other existing cells, and that the cell is a fundamental unit of structure, function and organization in all living organisms.


The cell was first discovered by Robert Hooke in 1665. He examined very thin slices of cork and saw billions of tiny pores that he remarked looked like the walled compartments of a honeycomb. Because of this association, Hooke called them cells, the name they still bear. However, Hooke did not know their real structure or function. [1] Hooke's description of these cells (which were actually non-living cell walls) was published in Micrographia.[2]. His cell observations gave no indication of the nucleus and other organelles found in most living cells.

The first man to witness a live cell under a microscope was Antonie van Leeuwenhoek, who in 1674 described the algae Spirogyra and named the moving organisms animalcules, meaning "little animals".[3]. Leeuwenhoek probably also saw bacteria.[4] Cell theory was in contrast to the vitalism theories that had been proposed before the discovery of cells.

The idea that cells were separable into individual units was proposed by Ludolph Christian Treviranus[5] and Johann Jacob Paul Moldenhawer[6]. All of this finally led to Henri Dutrochet formulating one of the fundamental tenets of modern cell theory by declaring that "The cell is the fundamental element of organization"[7]

The observations of Hooke, Leeuwenhoek, Schleiden, Schwann, Virchow, and others led to the development of the cell theory. The cell theory is a widely accepted explanation of the relationship between cells and living things. The cell theory states:

  1. All living things are composed of one or more cells.
  2. The cell is the most basic unit of life.
  3. All cells come from pre-existing cells.

The cell theory holds true for all living things, no matter how big or small, or how simple or complex. Since according to research, cells are common to all living things, they can provide information about all life. And because all cells come from other cells, scientists can study cells to learn about growth, reproduction, and all other functions that living things perform. By learning about cells and how they function, you can learn about all types of living things.Credit for developing cell theory is usually given to three scientists: Theodor Schwann, Matthias Jakob Schleiden, and Rudolf Virchow. In 1839, Schwann and Schleiden suggested that cells were the basic unit of life. Their theory accepted the first two tenets of modern cell theory (see next section, below). However the cell theory of Schleiden differed from modern cell theory in that it proposed a method of spontaneous crystallization that he called "Free Cell Formation"[8]. In 1858, Rudolf Virchow concluded that all cells come from pre-existing cells, thus completing the classical cell theory.

Classical interpretation

  1. All organisms are made up of one or more cells.
  2. Cells are the fundamental functional and structural unit of life.
  3. All cells come from pre-existing cells.
  4. The cell is the unit of structure, physiology, and organization in living things.
  5. The cell retains a dual existence as a distinct entity and a building block in the construction of organisms.

Modern interpretation

  1. The generally accepted parts of modern cell theory include:
  2. The cell is the fundamental unit of structure and function in living things.
  3. All cells come from pre-existing cells by division.
  4. Energy flow (metabolism and biochemistry) occurs within cells.
  5. Cells contain hereditary information (DNA) which is passed from cell to cell during cell division
  6. All cells are basically the same in chemical composition.
  7. All known living things are made up of cells.
  8. Some organisms are unicellular, i.e., made up of only one cell.
  9. Others are multicellular, composed of a number of cells.
  10. The activity of an organism depends on the total activity of independent cells.


  1. Viruses are considered by some to be alive, yet they are not made up of cells. Viruses have many of the features of life, but by definition of life, they are not alive.
  2. The first cell did not originate from a pre-existing cell. There was no exact first cell since the definition of cell is not that precise. This is an intellectual game that comes from making strict logical symbols out of the biological definitions.
  3. Mitochondria and chloroplasts have their own genetic material, and reproduce independently from the rest of the cell.

Types of cells

Cells can be subdivided into the following subcategories:

  1. Prokaryotes: Prokaryotes lack a nucleus (though they do have circular DNA) and other membrane-bound organelles (though they do contain ribosomes). Bacteria and Archaea are two divisions of prokaryotes.
  2. Eukaryotes: Eukaryotes, on the other hand, have distinct nuclei and membrane-bound organelles (mitochondria, chloroplasts, lysosomes, rough and smooth endoplasmic reticulum, vacuoles). In addition, they possess organized chromosomes which store genetic material.

Source : wikipedia.org


Algebra is a branch of mathematics concerning the study of structure, relation, and quantity. Together with geometry, analysis, combinatory, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.

Algebra is much broader than elementary algebra and can be generalized. In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.


While the word "algebra" comes from Arabic word (al-jabr , الجبر), its origins can be traced to the ancient Babylonians,[1] who developed an advanced arithmetical system with which they were able to do calculations in an algebraic fashion. With the use of this system they were able to apply formulas and calculate solutions for unknown values for a class of problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, and most Indian, Greek and Chinese mathematicians in the first millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Sulba Sutras, Euclid's Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations.

The Greek mathematicians Hero of Alexandria and Diaphanous [2] continued the traditions of Egypt and Babylon, but Diophantus's book Arithmetic is on a much higher level.[3] Later, Arab and Muslim mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khowarazmi was the first to solve equations using general methods. He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations and equations with multiple variable.

The word "algebra" is named after the Arabic word "al-jabr , الجبر" from the title of the book al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala , الكتاب المختصر في حساب الجبر والمقابلة, meaning The book of Summary Concerning Calculating by Transposition and Reduction, a book written by the Islamic Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī (considered the "father of algebra"), in 820. The word Al-Jabr means "reunion". The Hellenistic mathematician Diaphanous has traditionally been known as the "father of algebra" but in more recent times there is much debate over whether al-Khwarizmi, who founded the discipline of al-jabr, deserves that title instead.[4] Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical.[5] Those who support Al-Khwarizmi point to the fact that he introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al-jabr originally referred to,[6] and that he gave an exhaustive explanation of solving quadratic equations,[7] supported by geometric proofs, while treating algebra as an independent discipline in its own right.[8] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."[9]

The Persian mathematician Omar Khayyam developed algebraic geometry and found the general geometric solution of the cubic equation. Another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations.[10] He also developed the concept of a function.[11] The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji,[12] and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods.

Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Abstract algebra was developed in the 19th century, initially focusing on what is now called Galois theory, and on constructibility issues.


Algebra may be divided roughly into the following categories:

  • Elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra), also called second year and third year algebra;
  • Abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated.
  • Linear algebra, in which the specific properties of vector spaces are studied (including matrices);
  • Universal algebra, in which properties common to all algebraic structures are studied.
  • Algebraic number theory, in which the properties of numbers are studied through algebraic systems. Number theory inspired much of the original abstraction in algebra.
  • Algebraic geometry in its algebraic aspect.
  • Algebraic combinatorics, in which abstract algebraic methods are used to study combinatorial questions.

In some directions of advanced study, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a geometric structure (a metric or a topology) which is compatible with the algebraic structure. The list includes a number of areas of functional analysis:

  • Normed linear spaces
  • Banach spaces
  • Hilbert spaces
  • Banach algebras
  • Normed algebras
  • Topological algebras
  • Topological groups

Sumber : Wikipedia.org

Tuesday, March 03, 2009

Introduction to Mathematical Models in Biology

In the past, mathematics has not been as successful a tool in the biological science as it has been in physical science. There are probably many reasons for this. Plan and animal are complex, made up of many components. The simplest of which man is just learning to comprehend. There seems to be mo fundamental biological law analogous to Newton's Law. thus the scientific community is a long way from understanding cause effect relationship in the biological world to the same degree that such law exist for spring mass system. In addition, many animal possess an ability to choose course of action, an ability that cannot be attribute to a spring mass system nor a pendulum.

In the following sections we will develop mathematical models describing one aspect of the biological world, the mutual relationship among plants, animals, and their environment, a field of study known as ecology. As a means of quantifying this science, the number of individuals of different species is investigated.


the rate of change of the population as measured over the time interval would be

this indicates the absolute rate of increase of the population. A quantity which will prove to be quite important is the rate of change of the population per Individual, R(t). This called the growth rate per unit time as measured ove the interval

source by Richard Haberman "Mathematical Model"
to be continue

Monday, March 02, 2009


Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures.

A general example of a graph (actually, a pseudograph) with three vertices and six edges.

In the most common sense of the term a graph is an ordered pair G: = (V,E) comprising a set V of vertices or nodes together with a set E of edges or lines, which are 2-element subsets of V. To avoid ambiguity, this type of graph may be described precisely as undirected and simple.

Other senses of graph stem from different conceptions of the edge set. In one more generalized notion, E is a set together with a relation of incidence that associates with each edge two vertices. In another generalized notion, E is a multiset of unordered pairs of (not necessarily distinct) vertices. Many authors call this type of object a multigraph or pseudograph.

All of these variants and others are described more fully below.

The vertices belonging to an edge are called the ends, endpoints, or end vertices of the edge. A vertex may exist in a graph and not belong to an edge. V and E are usually taken to be finite, and many of the well-known results are not true (or are rather different) for infinite graphs because many of the arguments fail in the infinite case. The order of a graph is | V | (the number of vertices). A graph's size is | E | , the number of edges. The degree of a vertex is the number of edges that connect to it, where an edge that connects to the vertex at both ends (a loop) is counted twice.

The edges E of an undirected graph G induce a symmetric binary relation ~ on V that is called the adjacency relation of G. Specifically, for each edge {u,v} the vertices u and v are said to be adjacent to one another, which is denoted u ~ v. For an edge {u, v}, graph theorists usually use the somewhat shorter notation uv.

Types of graphs

Distinction in terms of the main definition

As stated above, in different contexts it may be useful to define the term graph with different degrees of generality. Whenever it is necessary to draw a strict distinction, the following terms are used. Most commonly, in modern texts in graph theory, unless stated otherwise, graph means "undirected simple finite graph" (see the definitions below).

Undirected graph

A graph in which edges have no orientation, i.e., they are not ordered pairs, but sets {u, v} (or 2-multisets) of vertices.

Directed graph

A directed graph or digraph is an ordered pair D: = (V,A) with

V a set whose elements are called vertices or nodes, and A a set of ordered pairs of vertices, called arcs, directed edges, or arrows. An arc a = (x,y) is considered to be directed from x to y; y is called the head and x is called the tail of the arc; y is said to be a direct successor of x, and x is said to be a direct predecessor of y. If a path leads from x to y, then y is said to be a successor of x and reachable from x, and x is said to be a predecessor of y. The arc (y,x) is called the arc (x,y) inverted.

A directed graph D is called symmetric if, for every arc in D, the corresponding inverted arc also belongs to D. A symmetric loopless directed graph D = (V, A) is equivalent to a simple undirected graph G = (V, E), where the pairs of inverse arcs in A correspond 1-to-1 with the edges in E; thus the edges in G number |E| = |A|/2, or half the number of arcs in D. A variation on this definition is the oriented graph, in which not more than one of (x,y) and (y,x) may be arcs.

Mixed graph

A mixed graph G is a graph in which some edges may be directed and some may be undirected. It is written as an ordered triple G := (V, E, A) with V, E, and A defined as above. Directed and undirected graphs are special cases.


A loop is an edge (directed or undirected) which starts and ends on the same vertex; these may be permitted or not permitted according to the application. In this context, an edge with two different ends is called a link. The term "multigraph" is generally understood to mean that multiple edges (and sometimes loops) are allowed. Where graphs are defined so as to allow loops and multiple edges, a multigraph is often defined to mean a graph without loops, however, where graphs are defined so as to disallow loops and multiple edges, the term is often defined to mean a "graph" which can have both multiple edges and loops, although many use the term "pseudograph" for this meaning.

Simple graph

A simple graph with three vertices and three edges. Each vertex has degree two, so this is also a regular graph.

As opposed to a multigraph, a simple graph is an undirected graph that has no loops and no more than one edge between any two different vertices. In a simple graph the edges of the graph form a set (rather than a multiset) and each edge is a pair of distinct vertices. In a simple graph with n vertices every vertex has a degree that is less than n (the converse, however, is not true - there exist non-simple graphs with n vertices in which every vertex has a degree smaller than n).

Weighted graph

A graph is a weighted graph if a number (weight) is assigned to each edge. Such weights might represent, for example, costs, lengths or capacities, etc. depending on the problem. The weight of the graph is sum of the weights given to all edges.

Regular graph

A regular graph is a graph where each vertex has the same number of neighbors, i.e., every vertex has the same degree or valency. A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k.

Complete graph

Complete graphs have the feature that each pair of vertices has an edge connecting them.

Finite and infinite graphs

A finite graph is a graph G = such that V(G) and E(G) are finite sets. An infinite graph is the one with sets of vertices or edges or both infinite. Most commonly in graph theory it is implied that the discussed graphs are finite, i.e., finite graphs are called simply "graphs", while the infinite graphs are called so in full.

In an undirected graph G, two vertices u and v are called connected if G contains a path from u to v. Otherwise, they are called disconnected. A graph is called connected if every pair of distinct vertices in the graph is connected and disconnected otherwise.

A graph is called k-vertex-connected or k-edge-connected if removal of k or more vertices (respectively, edges) makes the graph disconnected. A k-vertex-connected graph is often called simply k-connected.

A directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. It is strongly connected or strong if it contains a directed path from u to v and a directed path from v to u for every pair of vertices u,v.

Properties of graphs

Two edges of a graph are called adjacent (sometimes coincident) if they share a common vertex. Two arrows of a directed graph are called consecutive if the head of the first one is at the nock (notch end) of the second one. Similarly, two vertices are called adjacent if they share a common edge (consecutive if they are at the notch and at the head of an arrow), in which case the common edge is said to join the two vertices. An edge and a vertex on that edge are called incident.

The graph with only one vertex and no edges is called the trivial graph. A graph with only vertices and no edges is known as an edgeless graph. The graph with no vertices and no edges is sometimes called the null graph or empty graph, but not all mathematicians allow this object.

In a weighted graph or digraph, each edge is associated with some value, variously called its cost, weight, length or other term depending on the application; such graphs arise in many contexts, for example in optimal routing problems such as the traveling salesman problem.

Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may be called vertex-labeled. However, for many questions it is better to treat vertices as indistinguishable; then the graph may be called unlabeled. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges). The same remarks apply to edges, so that graphs which have labeled edges are called edge-labeled graphs. Graphs with labels attached to edges or vertices are more generally designated as labeled. Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called unlabeled. (Note that in the literature the term labeled may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges.)


A graph with six nodes.

The picture is a graphic representation of the following graph

V: = {1,2,3,4,5,6}

E: = {{1,2},{1,5},{2,3},{2,5},{3,4},{4,5},{4,6}}

The fact that vertex 1 is adjacent to vertex 2 is sometimes denoted by 1 ~ 2.

In category theory a category can be considered a directed multigraph with the objects as vertices and the morphisms as directed edges. The functors between categories induce then some, but not necessarily all, of the digraph morphisms. In computer science directed graphs are used to represent finite state machines and many other discrete structures.A binary relation R on a set X is a directed graph. Two edges x, y of X are connected by an arrow if xRy.

Important graphs

Basic examples are:

In a complete graph each pair of vertices is joined by an edge, that is, the graph contains all possible edges.

In a bipartite graph, the vertices can be divided into two sets, W and X, so that every edge has one vertex in each of the two sets.

In a complete bipartite graph, the vertex set is the union of two disjoint subsets, W and X, so that every vertex in W is adjacent to every vertex in X but there are no edges within W or X.

In a path of length n, the vertices can be listed in order, v0, v1, ..., vn, so that the edges are vi−1vi for each i = 1, 2, ..., n. A cycle or circuit of length n is a closed path without self-intersections; equivalently, it is a connected graph with degree 2 at every vertex. Its vertices can be named v1, ..., vn so that the edges are vi−1vi for each i = 2,...,n and vnv1

A planar graph can be drawn in a plane with no crossing edges (i.e., embedded in a plane).

A forest is a graph with no cycles.

A tree is a connected graph with no cycles.

More advanced kinds of graphs are:

The Petersen graph and its generalizations

Perfect graphs


Other graphs with large automorphism groups: vertex-transitive, arc-transitive, and distance-transitive graphs.

Strongly regular graphs and their generalization distance-regular graphs.

Operations on graphs

There are several operations that produce new graphs from old ones, which might be classified into the following categories: Elementary operations, sometimes called "editing operations" on graphs, which create a new graph from the original one by a simple, local change, such as addition or deletion of a vertex or an edge, merging and splitting of vertices, etc. Graph rewrite operations replacing the occurrence of some pattern graph within the host graph by an instance of the corresponding replacement graph.

Unary operations, which create a significantly new graph from the old one. Examples:

Line graph

Dual graph

Complement graph

Binary operations, which create new graph from two initial graphs. Examples:

Disjoint union of graphs

Cartesian product of graphs

Tensor product of graphs

Strong product of graphs

Lexicographic product of graphs

Source by wikipedia.org