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Fitness

Fitness


- Fitness in biology refers to individual's ability to propagate its genes.
- Fitness (optimization) in mathematics and computer science refers to the degree to which a given function is optimized or maximized.
- Physical fitness refers to a general state of good somatic health and abilities, usually as a result of exercise and nutrition.

Fitness (biology)

Insert non-formatted text here'Fitness (often denoted w in population genetics models) is a central concept in evolutionary theory. It describes the capability of an individual of certain genotype to reproduce, and usually is equal to the proportion of the individual's genes in all the genes of the next generation. If differences in individual genotypes affect fitness, then the frequencies of the genotypes will change over generations; the genotypes with higher fitness become more common. This process is called natural selection.

Measures of fitness

There are two commonly used measures of fitness; absolute fitness and relative fitness. Absolute fitness (w_) is defined for a given genotype for one generation as the ratio of organisms with that genotype after selection to those before. It may be calculated from absolute numbers or from frequencies, and as a ratio, is a value between 0 and 1. : = Relative fitness is quantified as the average number of surviving progeny of a particular genotype compared with average number of surviving progeny of competing genotypes after a single generation, i.e. one genotype is normalized at w=1 and the fitnesses of other genotypes are measured with respect to that genotype. Relative fitness can therefore take any positive value, including 0. While researchers can usually measure relative fitness, absolute fitness is more difficult. It is often difficult to determine how many individuals of a genotype there were immediately after reproduction. The two concepts are related, and both of them are equivalent when they are divided by the mean fitness, which is weighted by genotype frequencies. : = Because fitness is a coefficient, and that coefficient may be multiplied by several times, biologists may work with "log fitness" (particularly so before the advent of computers). By taking the logarithm of fitness each term may be added rather than multiplied.

Discussion

An individual's fitness is manifested through its phenotype. As phenotype is affected by both genes and environment, the fitnesses of different individuals with same genotype are not necessarily equal, but depend on the environment in which the individuals live. As fitness measures the quantity of the copies of the genes of an individual in the next generation, it doesn't really matter how the genes arrive in the next generation. That is, for an individual it is equally beneficial to reproduce itself, or to help relatives with similar genes to reproduce, as long as similar amount of copies of individual's genes get passed on to the next generation. Selection which promotes this kind of helper behaviour is called kin selection. A fitness landscape is a way of visualising fitness in terms of peaks, where natural selection will always push uphill but , resulting in suboptimality. Where there are differences in fitness, a genetic load is exerted on the population. Richard Dawkins introduced the controversial concept of ethical fitnessism.

History

The British economist Herbert Spencer coined the phrase "survival of the fittest" in his 1851 work Social Statics and later used it to characterise what Charles Darwin had called natural selection. The British biologist J.B.S. Haldane was the first to quantify fitness, in terms of the modern evolutionary synthesis of Darwinism and Mendelian genetics starting with his 1924 paper A Mathematical Theory of Natural and Artificial Selection. The next further advance was the introduction of the concept of inclusive fitness by the British biologist W.D. Hamilton in 1964 in his paper on The Evolution of Social Behavior.

References


- Haldane, J.B.S. (1924) "A mathematical theory of natural and artificial selection" Part 1 Transactions of the Camrbidge philosophical society: 23: 19-41 [http://www.blackwellpublishing.com/ridley/classictexts/haldane1.pdf link (pdf file)]
- Hamilton, W.D. (1964) "The evolution of social behavior" Journal of Theoretical Biology 1:...

External links


- [http://www.blackwellpublishing.com/ridley/a-z/Fitness.asp Evolution A-Z: Fitness]
- [http://plato.stanford.edu/entries/fitness/ Stanford Encyclopedia of Philosophy entry] Category:Evolutionary biologyCategory:Population genetics ja:適応度 '

Biology

Biology is the study, or science, of life. It is concerned with the characteristics and behaviors of organisms, how species and individuals come into existence, and the interactions they have with each other and with the environment. Biology encompasses a broad spectrum of academic fields that are often viewed as independent disciplines. However, together they address the phenomenon of life over a wide range of scales. At the atomic and molecular scale, life is studied in the disciplines of molecular biology, biochemistry, and molecular genetics. At the level of the cell, it is studied in cell biology, and at multicellular scales, it is examined in physiology, anatomy, and histology. Developmental biology studies life at the level of an individual organism's development or ontogeny. Moving up the scale towards more than one organism, genetics considers how heredity works between parent and offspring. Ethology considers group behavior of more than one individual. Population genetics looks at the level of an entire population, and systematics considers the multi-species scale of lineages. Interdependent populations and their habitats are examined in ecology and evolutionary biology. A speculative new field is astrobiology (or xenobiology), which examines the possibility of life beyond the Earth.
Biology studies the variety of life (clockwise from top-left) E. coli, tree fern, gazelle, Goliath beetle

Principles of biology

Unlike physics, biology does not usually describe systems in terms of objects which obey immutable physical laws described by mathematics. Nevertheless, the biological sciences are characterized and unified by several major underlying principles and concepts: universality, evolution, diversity, continuity, homeostasis, and interactions.

Universality: Biochemistry, cells, and the genetic code

mathematics]] Main articles: Life The most salient example of biological universality is that all living things share a common carbon-based biochemistry and in particular pass on their characteristics via genetic material, which is based on nucleic acids such as DNA and which uses a common genetic code with only minor variations. Another universal principle is that all organisms (that is, all forms of life on Earth except for viruses) are made of cells. Similarly, all organisms share common developmental processes. For example, in most metazoan organisms, the basic stages of early embryonic development share similar morphological characteristics and include similar genes.

Evolution: The central principle of biology

Main article: Evolution The central organizing concept in biology is that all life has a common origin and has changed and developed through the process of evolution (see Common descent). This has led to the striking similarity of units and processes discussed in the previous section. Charles Darwin established evolution as a viable theory by articulating its driving force, natural selection (Alfred Russell Wallace is recognized as the co-discoverer of this concept). Genetic drift was embraced as an additional mechanism of evolutionary development in the modern synthesis of the theory. The evolutionary history of a species— which describes the characteristics of the various species from which it descended— together with its genealogical relationship to every other species is called its phylogeny. Widely varied approaches to biology generate information about phylogeny. These include the comparisons of DNA sequences conducted within molecular biology or genomics, and comparisons of fossils or other records of ancient organisms in paleontology. Biologists organize and analyze evolutionary relationships through various methods, including phylogenetics, phenetics, and cladistics (The major events in the evolution of life, as biologists currently understand them, are summarized on this evolutionary timeline).

Diversity: The variety of living organisms

evolutionary timeline, based on rRNA gene data, showing the separation of the three domains bacteria, archaea, and eukaryotes as described initially by Carl Woese. Trees constructed with other genes are generally similar, although they may place some early-branching groups very differently, presumably owing to rapid rRNA evolution. The exact relationships of the three domains are still being debated.]] Despite its underlying unity, life exhibits an astonishingly wide diversity in morphology, behavior, and life histories. In order to grapple with this diversity, biologists attempt to classify all living things. Scientific classification seeks to reflect the evolutionary trees (phylogenetic trees) of the organism being classified. Classification is the province of the disciplines of systematics and taxonomy. Taxonomy places organisms in groups called taxa, while systematics seeks to define their relationships with each other. This clasification technique has evolved to reflect advances in cladistics and genetics, shifting the focus from physical similarities and shared characteristics to phylogenetics. Traditionally, living things have been divided into five kingdoms: :Monera -- Protista -- Fungi -- Plantae -- Animalia However, many scientists now consider this five-kingdom system to be outdated. Modern alternative classification systems generally begin with the three-domain system: :Archaea (originally Archaebacteria) -- Bacteria (originally Eubacteria) -- Eukaryota These domains reflect whether the cells have nuclei or not, as well as differences in the cell exteriors. There is also a series of intracellular parasites that are progressively "less alive" in terms of metabolic activity: :Viruses -- Viroids -- Prions

Continuity: The common descent of life

Main article: Common descent Up into the 19th century, it was commonly believed that life forms could appear spontaneously under certain conditions (see abiogenesis). This misconception was challenged by William Harvey's diction that "all life [is] from [an] egg" (from the Latin "Omne vivum ex ovo"), a foundational concept of modern biology. It simply means that there is an unbroken continuity of life from its initial origin to the present time. A group of organisms is said to share a common descent if they share a common ancestor. All organisms on the Earth have been and are descended from a common ancestor or an ancestral gene pool. This last universal common ancestor of all organisms is believed to have appeared about 3.5 billion years ago. Biologists generally regard the universality of the genetic code as definitive evidence in favor of the theory of universal common descent (UCD) for all bacteria, archaea, and eukaryotes (see: origin of life).

Homeostasis: Adapting to change

Main article: Homeostasis Homeostasis is the ability of an open system to regulate its internal environment to maintain a stable condition by means of multiple dynamic equilibrium adjustments controlled by interrelated regulation mechanisms. All living organisms, whether unicellular or multicellular, exhibit homeostasis. Homeostasis manifests itself at the cellular level through the maintenance of a stable internal acidity (pH); at the organismic level, warm-blooded animals maintain a constant internal body temperature; and at the level of the ecosystem, as when atmospheric carbon dioxide levels rise and plants are theoretically able to grow healthier and remove more of the gas from the atmosphere. Tissues and organs can also maintain homeostasis.

Interactions: Groups and environments

organ of the genus Amphiprion that dwell among the tentacles of tropical sea anemones. The territorial fish protects the anemone from anemone-eating fish, and in turn the stinging tentacles of the anemone protects the anemone fish from its predators]] Every living thing interacts with other organisms and its environment. One reason that biological systems can be difficult to study is that so many different interactions with other organisms and the environment are possible, even on the smallest of scales. A microscopic bacterium responding to a local sugar gradient is responding to its environment as much as a lion is responding to its environment when it searches for food in the African savannah. For any given species, behaviors can be co-operative, aggressive, parasitic or symbiotic. Matters become more complex when two or more different species interact in an ecosystem. Studies of this type are the province of ecology.

Scope of biology

Main article: List of biology disciplines Biology has become such a vast research enterprise that it is not generally regarded as a single discipline, but as a number of clustered sub-disciplines. This article considers four broad groupings. The first group consists of those disciplines that study the basic structures of living systems: cells, genes etc.; the second group considers the operation of these structures at the level of tissues, organs, and bodies; the third group considers organisms and their histories; the final constellation of disciplines focuses on their interactions. It is important to note, however, that these boundaries, groupings, and descriptions are a simplified characterization of biological research. In reality, the boundaries between disciplines are fluid, and most disciplines frequently borrow techniques from each other. For example, evolutionary biology leans heavily on techniques from molecular biology to determine DNA sequences, which assist in understanding the genetic variation of a population; and physiology borrows extensively from cell biology in describing the function of organ systems.

Structure of life

DNA sequences and structures]] Main articles: Molecular biology, Cell biology, Genetics, Developmental biology Molecular biology is the study of biology at a molecular level. This field overlaps with other areas of biology, particularly with genetics and biochemistry. Molecular biology chiefly concerns itself with understanding the interactions between the various systems of a cell, including the interrelationship of DNA, RNA, and protein synthesis and learning how these interactions are regulated. Cell biology studies the physiological properties of cells, as well as their behaviors, interactions, and environment. This is done both on a microscopic and molecular level. Cell biology researches both single-celled organisms like bacteria and specialized cells in multicellular organisms like humans. Understanding cell composition and how they function is fundamental to all of the biological sciences. Appreciating the similarities and differences between cell types is particularly important in the fields of cell and molecular biology. These fundamental similarities and differences provide a unifying theme, allowing the principles learned from studying one cell type to be extrapolated and generalized to other cell types. Genetics is the science of genes, heredity, and the variation of organisms. In modern research, genetics provides important tools in the investigation of the function of a particular gene, or the analysis of genetic interactions. Within organisms, genetic information generally is carried in chromosomes, where it is represented in the chemical structure of particular DNA molecules. Genes encode the information necessary for synthesizing proteins, which in turn play a large role in influencing (though, in many instances, not completely determining) the final phenotype of the organism. Developmental biology studies the process by which organisms grow and develop. Originating in embryology, modern developmental biology studies the genetic control of cell growth, differentiation, and "morphogenesis," which is the process that gives rise to tissues, organs, and anatomy. Model organisms for developmental biology include the round worm Caenorhabditis elegans, the fruit fly Drosophila melanogaster, the zebrafish Brachydanio rerio, the mouse Mus musculus, and the weed Arabidopsis thaliana.

Physiology of organisms

Main articles: Physiology, Anatomy Physiology studies the mechanical, physical, and biochemical processes of living organisms by attempting to understand how all of the structures function as a whole. The theme of "structure to function" is central to biology. Physiological studies have traditionally been divided into plant physiology and animal physiology, but the principles of physiology are universal, no matter what particular organism is being studied. For example, what is learned about the physiology of yeast cells can also apply to human cells. The field of animal physiology extends the tools and methods of human physiology to non-human species. Plant physiology also borrows techniques from both fields. Anatomy is an important branch of physiology and considers how organ systems in animals, such as the nervous, immune, endocrine, respiratory, and circulatory systems, function and interact. The study of these systems is shared with medically oriented disciplines such as neurology and immunology.

Diversity and evolution of organisms

immunology of a population of organisms is sometimes depicted as if travelling on a fitness landscape. The arrows indicate the preferred flow of a population on the landscape, and the points A, B, and C are local optima. The red ball indicates a population that moves from a very low fitness value to the top of a peak]] Main articles: Evolutionary biology, Botany, Zoology Evolutionary biology is concerned with the origin and descent of species, as well as their change over time, and includes scientists from many taxonomically-oriented disciplines. For example, it generally involves scientists who have special training in particular organisms such as mammalogy, ornithology, or herpetology, but use those organisms as systems to answer general questions about evolution. Evolutionary biology also makes use of paleontologists, who use the fossil record to answer questions about the mode and tempo of evolution, as well as theoreticians in areas such as population genetics and evolutionary theory. In the 1990s, developmental biology re-entered evolutionary biology from its initial exclusion from the modern synthesis through the study of evolutionary developmental biology. Related fields which are often considered part of evolutionary biology are phylogenetics, systematics, and taxonomy. The two major traditional taxonomically-oriented disciplines are botany and zoology. Botany is the scientific study of plants. Botany covers a wide range of scientific disciplines that study the growth, reproduction, metabolism, development, diseases, and evolution of plant life. Zoology involves the study of animals, including the study of their physiology within the fields of anatomy and embryology. The common genetic and developmental mechanisms of animals and plants is studied in molecular biology, molecular genetics, and developmental biology. The ecology of animals is covered under behavioral ecology and other fields.

Classification of life

The dominant classification system is called Linnaean taxonomy, which includes ranks and binomial nomenclature. How organisms are named is governed by international agreements such as the International Code of Botanical Nomenclature (ICBN), the International Code of Zoological Nomenclature (ICZN), and the International Code of Nomenclature of Bacteria (ICNB). A fourth Draft BioCode was published in 1997 in an attempt to standardize naming in these three areas, but it has yet to be formally adopted. The International Code of Virus Classification and Nomenclature (ICVCN) remains outside the BioCode.

Interactions of organisms

International Code of Virus Classification and Nomenclature]] Main articles: Ecology, Ethology, Behavior, Biogeography Ecology studies the distribution and abundance of living organisms, and the interactions between organisms and their environment. The environment of an organism includes both its habitat, which can be described as the sum of local abiotic factors such as climate and geology, as well as the other the organisms that share its habitat. Ecological systems are studied at several different levels, from individuals and populations to ecosystems and the biosphere. As can be surmised, ecology is a science that draws on several disciplines. Ethology studies animal behavior (particularly of social animals such as primates and canids), and is sometimes considered a branch of zoology. Ethologists have been particularly concerned with the evolution of behavior and the understanding of behavior in terms of the theory of natural selection. In one sense, the first modern ethologist was Charles Darwin, whose book The expression of the emotions in animals and men influenced many ethologists. Biogeography studies the spatial distribution of organisms on the Earth, focusing on topics like plate tectonics, climate change, dispersal and migration, and cladistics.

History of the word "biology"

Formed by combining the Greek βίος (bios), meaning 'life', and λόγος (logos), meaning 'study of', the word "biology" in its modern sense seems to have been introduced independently by Gottfried Reinhold Treviranus (Biologie oder Philosophie der lebenden Natur, 1802) and by Jean-Baptiste Lamarck (Hydrogéologie, 1802). The word itself is sometimes said to have been coined in 1800 by Karl Friedrich Burdach, but it appears in the title of Volume 3 of Michael Christoph Hanov's Philosophiae naturalis sive physicae dogmaticae: Geologia, biologia, phytologia generalis et dendrologia, published in 1766.

History

Main articles: History of biology, History of medicine, History of genetics Major discoveries in biology include:
- Cell theory
- Germ theory of disease
- Genetics
- Evolution
- DNA

Related topics

Main articles: List of biology topics

External links


- [http://www.rom.on.ca/biodiversity/biocode/biocode1997.html BioCode]: A proposal for organism naming.
- [http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=Books NCBI Open-Access Books]
- PhyloCode, [http://www.ohiou.edu/phylocode/index.html]
- [http://tolweb.org/tree/phylogeny.html The Tree of Life]: A multi-authored, distributed Internet project containing information about phylogeny and biodiversity.
- [http://www.bioone.org/perlserv/?request=index-html BioOne] Bioscience research journals.
- [http://www.bionews.in/biologynews.htm Biology News] Biology News, Articles and Research discoversies.

Further reading


- Lynn Margulis, Five Kingdoms: An Illustrated Guide to the Phyla of Life on Earth, 3rd ed., St. Martin's Press, 1997, paperback, ISBN 0805072527 (many other editions)
- Neil Campbell, Biology (7th edition), Benjamin-Cummings Publishing Company, 2004, hardcover, ISBN 080537146X
- Category:School subjects als:Biologie ko:생물학 ms:Biologi ja:生物学 simple:Biology th:ชีววิทยา

Gene

:For the musical band, see Gene (band) Gene (band) (right). Introns are regions often found in eukaryote genes which are removed in the splicing process: only the exons encode the protein. This diagram labels a region of only 40 or so bases as a gene. In reality many genes are much larger.]] Genes are regions of the DNA that parents pass to offspring during reproduction as chromosomes in nuclei of gametes. These entities encode information essential for the construction and regulation of proteins (such as enzymes) and other molecules that determine the growth and functioning of the organism. The word "gene" comes from the Greek genos ("origin") and is shared by many disciplines, including classical genetics, molecular genetics, evolutionary biology and population genetics. Because each discipline models the biology of life differently, the usage of the word gene varies between disciplines. It may refer to either material or conceptual entities. Following the discovery that DNA is the genetic material, and with the growth of biotechnology and the project to sequence the human genome, the common usage of the word "gene" has increasingly reflected its meaning in molecular biology. In the molecular-biological sense, genes are the segments of DNA which cells transcribe into RNA and translate, at least in part, into proteins. The [http://song.sourceforge.net/ Sequence Ontology] project defines a gene as: "A locatable region of genomic sequence, corresponding to a unit of inheritance, which is associated with regulatory regions, transcribed regions and/or other functional sequence regions". In common speech, "gene" is often used to refer to the hereditary cause of a trait, disease or condition—as in "the gene for obesity." Speaking more precisely, a biologist might refer to an allele or a mutation that has been implicated in or is associated with obesity. This is because biologists know that many factors other than genes decide whether a person is obese or not: eating habits, exercise, prenatal environment, upbringing, culture and the availability of food, for example. Moreover, it is very unlikely that variations within a single gene—or single genetic locus—fully determine one's genetic predisposition for obesity. These aspects of inheritance—the interplay between genes and environment, the influence of many genes—appear to be the norm with regard to many and perhaps most ("complex" or "multi-factoral") traits. The term phenotype refers to the characteristics that result from this interplay (see genotype-phenotype distinction).

Overview

Properties of genes

In molecular biology, the DNA of a gene encodes the chemical structure of a protein. The genetic code determines the sequence of the amino acids that make up a protein. The coding of a three nucleotide DNA sequence to a specific amino acid is essentially the same for all known life, from bacteria to humans. Through the proteins they encode, genes govern the cells in which they reside. In multicellular organisms they control the development of the individual from the fertilized egg and the day-to-day functions of the cells that make up tissues and organs. The instrumental roles of their protein products range from mechanical support of the cell structure to the transportation and manufacture of other molecules and to the regulation of other proteins' activities. The genes that exist today are those that have reproduced successfully in the past. Often, many individual organisms share a gene; thus, the death of an individual need not mean the extinction of the gene. Indeed, if the sacrifice of one individual enhances the survivability of other individuals with the same gene, the death of an individual may enhance the overall survival of the gene. This is the basis of the selfish gene view, popularized by Richard Dawkins. He points out in his book, The Selfish Gene, that to be successful genes need have no other "purpose" than to propagate themselves, even at the expense of their host organism's welfare. A human that behaved in such a way would be described as "selfish," although ironically a selfish gene may promote altruistic behaviours. According to Dawkins, the possibly disappointing answer to the question "what is the meaning of life?" may be "the survival and perpetuation of ribonucleic acids and their associated proteins".

Types of genes

Due to rare, spontaneous errors (e.g. in DNA replication) mutations in the sequence of a gene may arise. Once propagated to the next generation, this mutation may lead to variations within a species' population. Variants of a single gene are known as alleles, and differences in alleles may give rise to differences in traits, for example eye colour. A gene's most common allele is called the wild type allele, and rare alleles are called mutants. Normally, RNA is an intermediate product in the translation of a molecular gene into a protein. However, for some gene sequences, RNA molecules are actually the functional products. For example, RNAs known as ribozymes are capable of enzymatic function, or small interfering RNAs have a regulatory role. The DNA sequences from which such RNAs are transcribed are known as non-coding RNA, or RNA genes. All living organisms carry their genes and transmit them to offspring as DNA, but some viruses carry only RNA. Because they use RNA, their cellular hosts may synthesize their proteins as soon as they are infected and without the delay in waiting for transcription. On the other hand, RNA retroviruses, such as AIDS, require the reverse transcription of their genome from RNA into DNA before their proteins can be synthesized.

Human gene nomenclature

For each known human gene the HUGO Gene Nomenclature Committee (HGNC) approve a gene name and symbol (short-form abbreviation). All approved symbols are stored in the [http://www.gene.ucl.ac.uk/cgi-bin/nomenclature/searchgenes.pl HGNC Database]. Each symbol is unique and each gene is only given one approved gene symbol. It is necessary to provide a unique symbol for each gene so that people can talk about them. This also facilitates electronic data retrieval from publications. In preference each symbol maintains parallel construction in different members of a gene family and can be used in other species, especially the mouse.

Typical numbers of genes in an organism

The following table gives typical numbers of genes and genome size for some organisms. Estimates of the number of genes in an organism are somewhat controversial because they depend on the discovery of genes, and no techniques currently exist to prove that a DNA sequence contains no gene. (In early genetics, genes could be identified only if there were mutations, or alleles.) Nonetheless, estimates are made based on current knowledge.

Chemistry and function of genes

Chemical structure of a gene

Four kinds of sequentially linked nucleotides compose a DNA molecule or strand (more at DNA). These four nucleotides constitute the genetic alphabet. A sequence of three consecutive nucleotides, called a codon, is the protein-coding vocabulary. The sequence of codons in a gene specifies the amino-acid sequence of the protein it encodes. In most eukaryotic species, very little of the DNA in the genome encodes proteins, and the genes may be separated by vast sequences of so-called junk DNA. Moreover, the genes are often fragmented internally by non-coding sequences called introns, which can be many times longer than the genes themselves. Introns are removed on the heels of transcription by splicing. In the primary molecular sense, they represent parts of a gene, however. All the genes and intervening DNA together make up the genome of an organism, which in many species is divided among several chromosomes and typically present in two or more copies. The location (or locus) of a gene and the chromosome on which it is situated is in a sense arbitrary. Genes that appear together on the chromosomes of one species, such as humans, may appear on separate chromosomes in another species, such as mice. Two genes positioned near one another on a chromosome may encode proteins that figure in the same cellular process or in completely unrelated processes. As an example of the former, many of the genes involved in spermatogenesis reside together on the Y chromosome. Many species carry more than one copy of their genome within each of their somatic cells. These organisms are called diploid if they have two copies or polyploid if they have more than two copies. In such organisms, the copies are practically never identical. With respect to each gene, the copies that an individual possesses are liable to be distinct alleles, which may act synergistically or antagonistically to generate a trait or phenotype. The ways that gene copies interact are explained by chemical dominance relationships (more at genetics, allele).

Expression of molecular genes

For various reasons, the relationship between DNA strand and a phenotype trait is not direct. The same DNA strand in 2 different individuals may result in different traits because of the effect of other DNA strands or the environment.
- The DNA strand is expressed into a trait only if it is transcribed to RNA. Because the transcription starts from a specific base-pair sequence (a promoter) and stops at another (a terminator), our DNA strand needs to be correctly placed between the two. If not, it is considered as junk DNA, and is not expressed.
- Cells regulate the activity of genes in part by increasing or decreasing their rate of transcription. Over the short term, this regulation occurs through the binding or unbinding of proteins, known as transcription factors, to specific non-coding DNA sequences called regulatory elements. Therefore, to be expressed, our DNA strand needs to be properly regulated by other DNA strands.
- The DNA strand may also be silenced through DNA methylation or by chemical changes to the protein components of chromosomes (see histone). This is a permanent form of regulation of the transcription.
- The RNA is often edited before its translation into a protein. Eukaryotic cells splice the transcripts of a gene, by keeping the exons and removing the introns. Therefore, the DNA strand needs to be in an exon to be expressed. Because of the complexity of the splicing process, one transcribed RNA may be spliced in alternate ways to produce not one but a variety of proteins (alternative splicing) from one pre-mRNA. Prokaryotes produce a similar effect by shifting reading frames during translation.
- The translation of RNA into a protein also starts with a specific start and stop sequence.
- Once produced, the protein interacts with the many other proteins in the cell, according to the cell metabolism. This interaction finally produces the trait. This complex process helps explain the different meanings of "gene":
- a nucleotide sequence in a DNA strand;
- or the transcribed RNA, prior to splicing;
- or the transcribed RNA after splicing, i.e. without the introns The latter meaning of gene is the result of more "material entity" than the first one.

Mutations and evolution

Just as there are many factors influencing the expression of a particular DNA strand, there are many ways to have genetic mutations. For example, natural variations within regulatory sequences appear to underlie many of the heritable characteristics seen in organisms. The influence of such variations on the trajectory of evolution through natural selection may be as large as or larger than variation in sequences that encode proteins. Thus, though regulatory elements are often distinguished from genes in molecular biology, in effect they satisfy the shared and historical sense of the word. Indeed, a breeder or geneticist, in following the inheritance pattern of a trait, has no immediate way to know whether this pattern arises from coding sequences or regulatory sequences. Typically, he or she will simply attribute it to variations within a gene. Errors during DNA replication may lead to the duplication of a gene, which may diverge over time. Though the two sequences may remain the same, or be only slightly altered, they are typically regarded as separate genes (i.e. not as alleles of the same gene). The same is true when duplicate sequences appear in different species. Yet, though the alleles of a gene differ in sequence, nevertheless they are regarded as a single gene (occupying a single locus).

History

The existence of genes was first suggested by Gregor Mendel, who, in the 1860s, studied inheritance in pea plants and hypothesized a factor that conveys traits from parent to offspring. Although he did not use the term gene, he explained his results in terms of inherited characteristics. Mendel was also the first to hypothesize independent assortment, the distinction between dominant and recessive traits, the distinction between a heterozygote and homozygote, and the difference between what would later be described as genotype and phenotype. Mendel's concept was finally named when Wilhelm Johannsen coined the word gene in 1909. In the early 1900s, Mendel's work received renewed attention from scientists. In 1910, Thomas Hunt Morgan showed that genes reside on specific chromosomes. He later showed that genes occupy specific locations on the chromosome. With this knowledge, Morgan and his students began the first chromosomal map of the fruit fly Drosophila. In 1928, Frederick Griffith showed that genes could be transferred. In what is now known as Griffith's experiment, injections into a mouse of a deadly strain of bacteria that had been heat-killed transferred genetic information to a safe strain of the same bacteria, killing the mouse. In 1941, George Wells Beadle and Edward Lawrie Tatum showed that mutations in genes caused errors in certain steps in metabolic pathways. This showed that specific genes code for specific proteins, leading to the "one gene, one enzyme" hypothesis. Oswald Avery, Collin Macleod, and Maclyn McCarty showed in 1944 that DNA holds the gene's information. In 1953, James D. Watson and Francis Crick demonstrated the molecular structure of DNA. Together, these discoveries established the central dogma of molecular biology, which states that proteins are translated from RNA which is transcribed from DNA. This dogma has since been shown to have exceptions, such as reverse transcription in retroviruses.

Evolutionary concept of gene

George C. Williams first explicitly advocated the gene-centered view of evolution in his book Adaptation and Natural Selection. Also, he proposed an evolutionary concept of gene to be used when we are talking about natural selection favoring some gene. The definition is: ""that which segregates and recombines with appreciable frequency." Acording to this definition, even an asexual genome could be considered a gene, insofar it have an appreciable permanency through many generations. The difference is: the molecular gene transcribes as a unit, and the evolutionary gene inherits as a unit. Richard Dawkins' The Selfish Gene and The Extended Phenotype defended that the gene is the only replicator in livings systems. This means that only genes transmit their structure largely intact and are potentially immortal in the form of copies. So, genes should be the unit of selection.

See also


- DNA
- Gene-centered view of evolution
- Gene expression
- Gene therapy
- Gene family
- Genetic programming
- Genetic algorithm
- Genetics
- Genomes
- Genomes#Minimal genomes
- Genomics
- Homeobox
- Human Genome Project
- List of notable genes
- Meme
- Memetics
- Protein
- RNA

References

[http://print.google.com/print?id=WkHO9HI7koEC Google print]

External links


- [http://www.gene.ucl.ac.uk/nomenclature HUGO Gene Nomenclature Committee, HGNC]
- [http://www.gene.ucl.ac.uk/cgi-bin/nomenclature/searchgenes.pl the HGNC Database]
- [http://www.gene.ucl.ac.uk/hugo/ Human Genome Organisation, HUGO]
- [http://www.newscientist.com/news/news.jsp?id=ns99996561 Recount slashes number of human genes] (from New Scientist magazine)
  - [http://www.genome.gov/12513430 National Human Genome Research Institute — News Release]
  - [http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v431/n7011/full/nature03001_fs.html Nature - 21 October 2004 — Finishing the euchromatic sequence of the human genome]
- [http://www.ncbi.nlm.nih.gov/mapview/stats/BuildStats.cgi?taxid=10116&build=3&ver=1#contigs Rat Genome]
- [http://plato.stanford.edu/entries/gene/ Stanford Encyclopedia of Philosophy entry]
- [http://www.ihop-net.org/UniPub/iHOP/ iHOP - Information Hyperlinked over Proteins]
- [http://www.pir.uniprot.org/ UniProt] Category:Cloning Category:Genetics Category:Molecular biology ko:유전자 ja:遺伝子 simple:Gene th:หน่วยพันธุกรรม

Optimization

In mathematics, optimization is the discipline which is concerned with finding the maxima and minima of functions, possibly subject to constraints. An example of an optimization problem is the following: maximize the profit of a manufacturing operation while ensuring that none of the resources exceed certain limits and also satisfying as much of the demand faced as possible. Optimization has many practical applications in logistics and design problems. In computer science, optimization is the process of improving a system in certain ways to increase the effective execution speed and/or bandwidth, or to reduce memory requirements. Despite its name, optimization does not necessarily mean finding the optimum solution to a problem. Often this is not possible, and heuristic algorithms must be used instead. Also in computer science, search engine optimization (SEO) is a set of methodologies aimed at improving the ranking of a website in search engine listings. The term also refers to an industry of consultants that carry out optimization projects on behalf of client sites. ----

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Quantity

This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. : :Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

:Pinning down ideas of size, symmetry, and mathematical structure. : :Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Space

:A more visual approach to mathematics. : :Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

See also list of mathematics history topics :History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

See also


- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle

Bibliography


- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links


- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Function (mathematics)

In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). The concept of a function is fundamental to virtually every branch of mathematics and every quantitative science. The terms function, mapping, map and transformation are usually used synonymously. The term operation is frequently used for binary functions; functions whose domain is a set of functions, or a vector space, are often called operators (see also operator (programming)).

Intuitive introduction

Essentially, a function is a "rule" or procedure that assigns an "output" value to each given "input" value. The following are examples of functions:
- In a group of people, each person has a favorite colour—from the set of red, orange, yellow, green, cyan, blue, indigo, or violet. Here, the input is the person, and the output is one of the 8 colours. The favorite colour is a function of the person. For example, John has favorite colour red, while Kim has favorite colour violet. Note that more than one person may be associated with a given colour (e.g., John and Kim may both like red), but one person cannot have more or less than one favorite color.
- A stone is dropped from different stories of a tall building. The dropped stone may take 2 seconds to fall from the second story, and 4 seconds to fall from the 8th story. Here, the input is the story, and the output is the number of seconds. The relevant function describes the relationship between the time it takes the stone to reach the ground and the story. (See acceleration) The "rule" defining a function can be specified by a formula, a relationship, or simply a table listing the outputs against inputs. The most important feature of a function is that it is consistent, or deterministic, always producing the same output from a given input. In this way, a function may be thought of as a mechanism or "machine" (a "black box") consistently converting a given valid input into its unique associated output. In certain technical contexts, the input is often called the argument of the function, and the output the value of the function. A very common type of function occurs when the argument (input) and the value (output) are both numbers, the functional relationship is expressed by a formula, and the value (output) of the function is obtained by direct substitution of the argument into the formula. Consider for example :f(x)=x^ which for any number x, assigns to x the associated value the square of x. A straightforward generalization is to allow functions depending on several arguments. For instance, :g(x,y) = xy is a function which takes the input, two expressions x and y, and assigns to it its product (output), xy. It might seem that this is not really a function as we described above, because this "rule" depends on two inputs. However, if we think of the two inputs together as a single pair (x, y), then we can interpret g as a function -- the argument (unified single input) is the ordered pair (x, y), and the function value (output) is xy. Such functions whose input consists of ordered pairs are called "binary" or "2-ary". In the sciences, we often encounter functions that are not given by (known) formulas. Consider for instance the temperature distribution on earth over time: this is a function which takes location and time as arguments and gives as output value the temperature at the indicated location at the indicated moment in time. We have seen that the intuitive notion of function is not limited to computations using single numbers and not even limited to computations; the mathematical notion of function is still more general and is not limited to situations involving numbers. Rather, a function links a "domain" (set of inputs) to a "codomain" (set of possible outputs) in such a way that every element of the domain is associated to precisely one element of the codomain. Functions are abstractly defined as certain relations, as will be seen below. Because of this generality, the function concept is fundamental to virtually every branch of mathematics and the quantitative sciences.

History

As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope or a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the change of output values associated to a change of input values, and these measurements are the basis of calculus. The word function was later used by Euler during the mid-18th century to describe an expression or formula involving various arguments, e.g. f(x) = sin(x) + x3. During the 19th century, mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on arithmetic rather than on