1.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
2.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
3.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
4.
Physical cosmology
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Physical cosmology is the study of the largest-scale structures and dynamics of the Universe and is concerned with fundamental questions about its origin, structure, evolution, and ultimate fate. These advances made it possible to speculate about the origin of the universe, a few researchers still advocate a handful of alternative cosmologies, however, most cosmologists agree that the Big Bang theory explains the observations better. Cosmology draws heavily on the work of many areas of research in theoretical. Areas relevant to include particle physics experiments and theory, theoretical and observational astrophysics, general relativity, quantum mechanics. Modern cosmology developed along tandem tracks of theory and observation, in 1916, Albert Einstein published his theory of general relativity, which provided a unified description of gravity as a geometric property of space and time. At the time, Einstein believed in a universe. This is because masses distributed throughout the universe gravitationally attract, however, he realized that his equations permitted the introduction of a constant term which could counteract the attractive force of gravity on the cosmic scale. Einstein published his first paper on relativistic cosmology in 1917, in which he added this cosmological constant to his equations in order to force them to model a static universe. However, this so-called Einstein model is unstable to small perturbations—it will eventually start to expand or contract, the Einstein model describes a static universe, space is finite and unbounded. It was later realized that Einsteins model was just one of a set of possibilities, all of which were consistent with general relativity. The cosmological solutions of general relativity were found by Alexander Friedmann in the early 1920s and his equations describe the Friedmann–Lemaître–Robertson–Walker universe, which may expand or contract, and whose geometry may be open, flat, or closed. In the 1910s, Vesto Slipher interpreted the red shift of spiral nebulae as a Doppler shift that indicated they were receding from Earth, however, it is difficult to determine the distance to astronomical objects. One way is to compare the size of an object to its angular size. Another method is to measure the brightness of an object and assume an intrinsic luminosity, due to the difficulty of using these methods, they did not realize that the nebulae were actually galaxies outside our own Milky Way, nor did they speculate about the cosmological implications. In 1929, Edwin Hubble provided a basis for Lemaîtres theory. Hubble showed that the nebulae were galaxies by determining their distances using measurements of the brightness of Cepheid variable stars. He discovered a relationship between the redshift of a galaxy and its distance and he interpreted this as evidence that the galaxies are receding from Earth in every direction at speeds proportional to their distance. Given the cosmological principle, Hubbles law suggested that the universe was expanding, two primary explanations were proposed for the expansion
5.
Cryptography
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Cryptography or cryptology is the practice and study of techniques for secure communication in the presence of third parties called adversaries. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, Applications of cryptography include ATM cards, computer passwords, and electronic commerce. Cryptography prior to the age was effectively synonymous with encryption. The originator of an encrypted message shared the decoding technique needed to recover the information only with intended recipients. The cryptography literature often uses Alice for the sender, Bob for the intended recipient and it is theoretically possible to break such a system, but it is infeasible to do so by any known practical means. The growth of technology has raised a number of legal issues in the information age. Cryptographys potential for use as a tool for espionage and sedition has led governments to classify it as a weapon and to limit or even prohibit its use. In some jurisdictions where the use of cryptography is legal, laws permit investigators to compel the disclosure of encryption keys for documents relevant to an investigation, Cryptography also plays a major role in digital rights management and copyright infringement of digital media. Until modern times, cryptography referred almost exclusively to encryption, which is the process of converting ordinary information into unintelligible text, decryption is the reverse, in other words, moving from the unintelligible ciphertext back to plaintext. A cipher is a pair of algorithms that create the encryption, the detailed operation of a cipher is controlled both by the algorithm and in each instance by a key. The key is a secret, usually a short string of characters, historically, ciphers were often used directly for encryption or decryption without additional procedures such as authentication or integrity checks. There are two kinds of cryptosystems, symmetric and asymmetric, in symmetric systems the same key is used to encrypt and decrypt a message. Data manipulation in symmetric systems is faster than asymmetric systems as they generally use shorter key lengths, asymmetric systems use a public key to encrypt a message and a private key to decrypt it. Use of asymmetric systems enhances the security of communication, examples of asymmetric systems include RSA, and ECC. Symmetric models include the commonly used AES which replaced the older DES, in colloquial use, the term code is often used to mean any method of encryption or concealment of meaning. However, in cryptography, code has a specific meaning. It means the replacement of a unit of plaintext with a code word, English is more flexible than several other languages in which cryptology is always used in the second sense above. RFC2828 advises that steganography is sometimes included in cryptology, the study of characteristics of languages that have some application in cryptography or cryptology is called cryptolinguistics
6.
Scientific notation
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Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is commonly used by scientists, mathematicians and engineers, in part because it can simplify certain arithmetic operations, on scientific calculators it is known as SCI display mode. In scientific notation all numbers are written in the form m × 10n, where the exponent n is an integer, however, the term mantissa may cause confusion because it is the name of the fractional part of the common logarithm. If the number is then a minus sign precedes m. In normalized notation, the exponent is chosen so that the value of the coefficient is at least one. Decimal floating point is an arithmetic system closely related to scientific notation. Any given integer can be written in the form m×10^n in many ways, in normalized scientific notation, the exponent n is chosen so that the absolute value of m remains at least one but less than ten. Thus 350 is written as 3. 5×102 and this form allows easy comparison of numbers, as the exponent n gives the numbers order of magnitude. In normalized notation, the exponent n is negative for a number with absolute value between 0 and 1, the 10 and exponent are often omitted when the exponent is 0. Normalized scientific form is the form of expression of large numbers in many fields, unless an unnormalized form. Normalized scientific notation is often called exponential notation—although the latter term is general and also applies when m is not restricted to the range 1 to 10. Engineering notation differs from normalized scientific notation in that the exponent n is restricted to multiples of 3, consequently, the absolute value of m is in the range 1 ≤ |m| <1000, rather than 1 ≤ |m| <10. Though similar in concept, engineering notation is rarely called scientific notation, engineering notation allows the numbers to explicitly match their corresponding SI prefixes, which facilitates reading and oral communication. A significant figure is a digit in a number that adds to its precision and this includes all nonzero numbers, zeroes between significant digits, and zeroes indicated to be significant. Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore,1,230,400 usually has five significant figures,1,2,3,0, and 4, when a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but the place holding zeroes are no longer required, thus 1,230,400 would become 1.2304 ×106. However, there is also the possibility that the number may be known to six or more significant figures, thus, an additional advantage of scientific notation is that the number of significant figures is clearer
7.
Logarithmic scale
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A logarithmic scale is a nonlinear scale used when there is a large range of quantities. Common uses include the strength, sound loudness, light intensity. It is based on orders of magnitude, rather than a linear scale. In particular our sense of hearing perceives equal ratios of frequencies as equal differences in pitch, the top left graph is linear in the X and Y axis, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y axis of the left graph. The top right graph uses a scale for just the X axis. A slide rule has logarithmic scales, and nomograms often employ logarithmic scales, the geometric mean of two numbers is midway between the numbers. Before the advent of graphics, logarithmic graph paper was a commonly used scientific tool. If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a log–log plot, if only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi-logarithmic plot. Bit Byte Decade John Napier Level Logarithm Logarithmic mean Preferred number Dehaene, Stanislas, Izard, Véronique, Spelke, Elizabeth, Pica, distinct intuitions of the number scale in Western and Amazonian indigene cultures. American Association for the Advancement of Science, why using logarithmic scale to display share prices
8.
1,000,000,000
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1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001. One billion can also be written as b or bn, in scientific notation, it is written as 1 ×109. The SI prefix giga indicates 1,000,000,000 times the base unit, one billion years may be called eon in astronomy and geology. Previously in British English, the word billion referred exclusively to a million millions, however, this is no longer as common as earlier, and the word has been used to mean one thousand million for some time. The alternative term one thousand million is used in the U. K. or countries such as Spain that uses one thousand million as one million million constitutes a billion. The worded figure, as opposed to the figure is used to differentiate between one thousand million or one billion. The term milliard can also be used to refer to 1,000,000,000, whereas milliard is seldom used in English, in the South Asian numbering system, it is known as 100 crore or 1 Arab. 1000000007 – smallest prime number with 10 digits,1023456789 – smallest pandigital number in base 10. 1026753849 – smallest pandigital square that includes 0,1073741824 –2301073807359 – 14th Kynea number. 1162261467 –3191220703125 –513 1232922769- 35113^2 Centered hexagonal number,1234567890 – pandigital number with the digits in order. 1882341361 – The least prime whose reversal is both square and triangular,1977326743 –7112147483647 – 8th Mersenne prime and the largest signed 32-bit integer. 2147483648 –2312176782336 –6122214502422 – 6th primary pseudoperfect number,2357947691 –1192971215073 – 11th Fibonacci prime. 3405691582 – hexadecimal CAFEBABE, used as a placeholder in programming,3405697037 – hexadecimal CAFED00D, used as a placeholder in programming. 3735928559 – hexadecimal DEADBEEF, used as a placeholder in programming,3486784401 –3204294836223 – 16th Carol number. 4294967291 – Largest prime 32-bit unsigned integer,4294967295 – Maximum 32-bit unsigned integer, perfect totient number, product of the five prime Fermat numbers. 4294967296 –2324294967297 – the first composite Fermat number,6103515625 –5146210001000 – only self-descriptive number in base 10. 6975757441 –1786983776800 – 15th colossally abundant number, 15th superior highly composite number 7645370045 – 27th Pell number,8589934592 –2339043402501 – 25th Motzkin number. 9814072356 – largest square pandigital number, largest pandigital pure power,9876543210 – largest number without redundant digits
9.
Hard disk drive
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The platters are paired with magnetic heads, usually arranged on a moving actuator arm, which read and write data to the platter surfaces. Data is accessed in a manner, meaning that individual blocks of data can be stored or retrieved in any order. HDDs are a type of storage, retaining stored data even when powered off. Introduced by IBM in 1956, HDDs became the dominant secondary storage device for computers by the early 1960s. Continuously improved, HDDs have maintained this position into the era of servers. More than 200 companies have produced HDDs historically, though after extensive industry consolidation most current units are manufactured by Seagate, Toshiba, as of 2016, HDD production is growing, although unit shipments and sales revenues are declining. While SSDs have higher cost per bit, SSDs are replacing HDDs where speed, power consumption, small size, the primary characteristics of an HDD are its capacity and performance. Capacity is specified in unit prefixes corresponding to powers of 1000, the two most common form factors for modern HDDs are 3. 5-inch, for desktop computers, and 2. 5-inch, primarily for laptops. HDDs are connected to systems by standard interface cables such as PATA, SATA, Hard disk drives were introduced in 1956, as data storage for an IBM real-time transaction processing computer and were developed for use with general-purpose mainframe and minicomputers. The first IBM drive, the 350 RAMAC in 1956, was approximately the size of two medium-sized refrigerators and stored five million six-bit characters on a stack of 50 disks. In 1962 the IBM350 RAMAC disk storage unit was superseded by the IBM1301 disk storage unit, cylinder-mode read/write operations were supported, and the heads flew about 250 micro-inches above the platter surface. Motion of the head array depended upon a binary system of hydraulic actuators which assured repeatable positioning. The 1301 cabinet was about the size of three home refrigerators placed side by side, storing the equivalent of about 21 million eight-bit bytes, access time was about a quarter of a second. Also in 1962, IBM introduced the model 1311 disk drive, users could buy additional packs and interchange them as needed, much like reels of magnetic tape. Later models of removable pack drives, from IBM and others, became the norm in most computer installations, non-removable HDDs were called fixed disk drives. Some high-performance HDDs were manufactured with one head per track so that no time was lost physically moving the heads to a track, known as fixed-head or head-per-track disk drives they were very expensive and are no longer in production. In 1973, IBM introduced a new type of HDD code-named Winchester and its primary distinguishing feature was that the disk heads were not withdrawn completely from the stack of disk platters when the drive was powered down. Instead, the heads were allowed to land on an area of the disk surface upon spin-down
10.
Gigabyte
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The gigabyte is a multiple of the unit byte for digital information. The prefix giga means 109 in the International System of Units, the unit symbol for the gigabyte is GB. However, the term is used in some fields of computer science and information technology to denote 1073741824 bytes. The use of gigabyte may thus be ambiguous, to address this ambiguity, the International System of Quantities standardizes the binary prefixes which denote a series of integer powers of 1024. With these prefixes, a module that is labeled as having the size 1GB has one gibibyte of storage capacity. The term gigabyte is commonly used to mean either 10003 bytes or 10243 bytes, the latter binary usage originated as compromise technical jargon for byte multiples that needed to be expressed in a power of 2, but lacked a convenient name. As 1024 is approximately 1000, roughly corresponding to SI multiples, in 1998 the International Electrotechnical Commission published standards for binary prefixes, requiring that the gigabyte strictly denote 10003 bytes and gibibyte denote 10243 bytes. By the end of 2007, the IEC Standard had been adopted by the IEEE, EU, and NIST and this is the recommended definition by the International Electrotechnical Commission. The file manager of Mac OS X version 10.6 and later versions are an example of this usage in software. The binary definition uses powers of the base 2, as is the principle of binary computers. This usage is widely promulgated by some operating systems, such as Microsoft Windows in reference to computer memory and this definition is synonymous with the unambiguous unit gibibyte. Since the first disk drive, the IBM350, disk drive manufacturers expressed hard drive capacities using decimal prefixes, with the advent of gigabyte-range drive capacities, manufacturers based most consumer hard drive capacities in certain size classes expressed in decimal gigabytes, such as 500 GB. The exact capacity of a given model is usually slightly larger than the class designation. Practically all manufacturers of disk drives and flash-memory disk devices continue to define one gigabyte as 1000000000bytes. Some operating systems such as OS X express hard drive capacity or file size using decimal multipliers and this discrepancy causes confusion, as a disk with an advertised capacity of, for example,400 GB might be reported by the operating system as 372 GB, meaning 372 GiB. The JEDEC memory standards use IEEE100 nomenclature which quote the gigabyte as 1073741824bytes and this means that a 300 GB hard disk might be indicated variously as 300 GB,279 GB or 279 GiB, depending on the operating system. As storage sizes increase and larger units are used, these differences even more pronounced. Some legal challenges have been waged over this confusion such as a lawsuit against drive manufacturer Western Digital, Western Digital settled the challenge and added explicit disclaimers to products that the usable capacity may differ from the advertised capacity
11.
Atom
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An atom is the smallest constituent unit of ordinary matter that has the properties of a chemical element. Every solid, liquid, gas, and plasma is composed of neutral or ionized atoms, Atoms are very small, typical sizes are around 100 picometers. Atoms are small enough that attempting to predict their behavior using classical physics - as if they were billiard balls, through the development of physics, atomic models have incorporated quantum principles to better explain and predict the behavior. Every atom is composed of a nucleus and one or more bound to the nucleus. The nucleus is made of one or more protons and typically a number of neutrons. Protons and neutrons are called nucleons, more than 99. 94% of an atoms mass is in the nucleus. The protons have an electric charge, the electrons have a negative electric charge. If the number of protons and electrons are equal, that atom is electrically neutral, if an atom has more or fewer electrons than protons, then it has an overall negative or positive charge, respectively, and it is called an ion. The electrons of an atom are attracted to the protons in a nucleus by this electromagnetic force. The number of protons in the nucleus defines to what chemical element the atom belongs, for example, the number of neutrons defines the isotope of the element. The number of influences the magnetic properties of an atom. Atoms can attach to one or more other atoms by chemical bonds to form compounds such as molecules. The ability of atoms to associate and dissociate is responsible for most of the changes observed in nature. The idea that matter is made up of units is a very old idea, appearing in many ancient cultures such as Greece. The word atom was coined by ancient Greek philosophers, however, these ideas were founded in philosophical and theological reasoning rather than evidence and experimentation. As a result, their views on what look like. They also could not convince everybody, so atomism was but one of a number of competing theories on the nature of matter. It was not until the 19th century that the idea was embraced and refined by scientists, in the early 1800s, John Dalton used the concept of atoms to explain why elements always react in ratios of small whole numbers
12.
Nucleon
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In chemistry and physics, nucleons are the proton and the neutron, which are the constituents of the atomic nucleus, and are themselves composed of down and up quarks. The number of nucleons defines an isotopes mass number, the atom is made up of a nucleus surrounded by electrons. Until the 1960s, nucleons were thought to be elementary particles, now they are known to be composite particles, made of three quarks bound together by the so-called strong interaction. The interaction between two or more nucleons is called internucleon interactions or nuclear force, which is ultimately caused by the strong interaction. Nucleons sit at the boundary where particle physics and nuclear physics overlap, Particle physics, particularly quantum chromodynamics, provides the fundamental equations that explain the properties of quarks and of the strong interaction. These equations explain quantitatively how quarks can bind together into protons and neutrons, however, when multiple nucleons are assembled into an atomic nucleus, these fundamental equations become too difficult to solve directly. Instead, nuclides are studied within nuclear physics, which studies nucleons and their interactions by approximations and models and these models can successfully explain nuclide properties, for example, whether or not a certain nuclide undergoes radioactive decay. The proton and neutron are both baryons and both fermions, one carries a positive net charge and the other carries a zero net charge, the protons mass is only 0. 1% less than the neutrons. Thus, they can be viewed as two states of the nucleon, and together form an isospin doublet. In isospin space, neutrons can be transformed into protons via SU symmetries and these nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. According to the Noether theorem, isospin is conserved with respect to the strong interaction, protons and neutrons are most important and best known for constituting atomic nuclei, but they can also be found on their own, not part of a larger nucleus. A proton on its own is the nucleus of the hydrogen-1 atom, a neutron on its own is unstable, but they can be found in nuclear reactions and are used in scientific analysis. Both the proton and neutron are made of three quarks, the proton is made of two up quarks and one down quark, while the neutron is one up quark and two down quarks. The quarks are held together by the force, or equivalently, by gluons. An up quark has electric charge + 2⁄3 e, and a down quark has charge − 1⁄3 e, so the electric charge of the proton and neutron are +e and 0. The word neutron comes from the fact that it is electrically neutral, the mass of the proton and neutron is quite similar, The proton is 1. 6726×10−27 kg or 938.27 MeV/c2, while the neutron is 1. 6749×10−27 kg or 939.57 MeV/c2. The neutron is roughly 0. 1% heavier, the similarity in mass can be explained roughly by the slight difference in mass of up quark and down quark composing the nucleons. However, detailed explanation remains a problem in particle physics
13.
Cell (biology)
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The cell is the basic structural, functional, and biological unit of all known living organisms. A cell is the smallest unit of life that can replicate independently, the study of cells is called cell biology. Cells consist of cytoplasm enclosed within a membrane, which contains many such as proteins. Organisms can be classified as unicellular or multicellular, while the number of cells in plants and animals varies from species to species, humans contain more than 10 trillion cells. Most plant and animal cells are only under a microscope. The cell was discovered by Robert Hooke in 1665, who named the unit for its resemblance to cells inhabited by Christian monks in a monastery. Cells emerged on Earth at least 3.5 billion years ago, Cells are of two types, eukaryotic, which contain a nucleus, and prokaryotic, which do not. Prokaryotes are single-celled organisms, while eukaryotes can be either single-celled or multicellular, prokaryotic cells were the first form of life on Earth, characterised by having vital biological processes including cell signaling and being self-sustaining. They are simpler and smaller than eukaryotic cells, and lack membrane-bound organelles such as the nucleus, prokaryotes include two of the domains of life, bacteria and archaea. The DNA of a prokaryotic cell consists of a chromosome that is in direct contact with the cytoplasm. The nuclear region in the cytoplasm is called the nucleoid, most prokaryotes are the smallest of all organisms ranging from 0.5 to 2.0 µm in diameter. Though most prokaryotes have both a cell membrane and a wall, there are exceptions such as Mycoplasma and Thermoplasma which only possess the cell membrane layer. The envelope gives rigidity to the cell and separates the interior of the cell from its environment, the cell wall consists of peptidoglycan in bacteria, and acts as an additional barrier against exterior forces. It also prevents the cell from expanding and bursting from osmotic pressure due to a hypotonic environment, some eukaryotic cells also have a cell wall. Inside the cell is the region that contains the genome, ribosomes. The genetic material is found in the cytoplasm. Prokaryotes can carry extrachromosomal DNA elements called plasmids, which are usually circular, linear bacterial plasmids have been identified in several species of spirochete bacteria, including members of the genus Borrelia notably Borrelia burgdorferi, which causes Lyme disease. Though not forming a nucleus, the DNA is condensed in a nucleoid, plasmids encode additional genes, such as antibiotic resistance genes
14.
Shannon number
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Shannon also estimated the number of possible positions, of the general order of 64. This includes some illegal positions and excludes legal positions following captures, taking these into account, Victor Allis calculated an upper bound of 5×1052 for the number of positions, and estimated the true number to be about 1050. Recent results improve that estimate, by proving an upper bound of only 2155, Allis also estimated the game-tree complexity to be at least 10123, based on an average branching factor of 35 and an average game length of 80. As a comparison, the number of atoms in the observable universe, as a comparison to the Shannon number, if chess is analyzed for the number of sensible games that can be played, then the result is closer to around 1040 games. This is based on having a choice of about three moves at each ply, and a game length of 80 ply. Solving chess Go and mathematics Mathematics and chess
15.
Avogadro constant
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In chemistry and physics, the Avogadro constant is the number of constituent particles, usually atoms or molecules, that are contained in the amount of substance given by one mole. Thus, it is the proportionality factor that relates the mass of a compound to the mass of a sample. Avogadros constant, often designated with the symbol NA or L, has the value 7023602214085700000♠6. 022140857×1023 mol−1 in the International System of Units and this number is also known as Loschmidt constant in German literature. The constant was later redefined as the number of atoms in 12 grams of the isotope carbon-12, for instance, to a first approximation,1 gram of hydrogen element, having the atomic number 1, has 7023602200000000000♠6. 022×1023 hydrogen atoms. Similarly,12 grams of 12C, with the mass number 12, has the number of carbon atoms. Avogadros number is a quantity, and has the same numerical value of the Avogadro constant given in base units. In contrast, the Avogadro constant has the dimension of reciprocal amount of substance, the Avogadro constant can also be expressed as 0.602214. ML mol−1 Å−3, which can be used to convert from volume per molecule in cubic ångströms to molar volume in millilitres per mole, revisions in the base set of SI units necessitated redefinitions of the concepts of chemical quantity. Avogadros number, and its definition, was deprecated in favor of the Avogadro constant, the French physicist Jean Perrin in 1909 proposed naming the constant in honor of Avogadro. Perrin won the 1926 Nobel Prize in Physics, largely for his work in determining the Avogadro constant by several different methods, accurate determinations of Avogadros number require the measurement of a single quantity on both the atomic and macroscopic scales using the same unit of measurement. This became possible for the first time when American physicist Robert Millikan measured the charge on an electron in 1910, the electric charge per mole of electrons is a constant called the Faraday constant and had been known since 1834 when Michael Faraday published his works on electrolysis. By dividing the charge on a mole of electrons by the charge on a single electron the value of Avogadros number is obtained, since 1910, newer calculations have more accurately determined the values for the Faraday constant and the elementary charge. Perrin originally proposed the name Avogadros number to refer to the number of molecules in one gram-molecule of oxygen, with this recognition, the Avogadro constant was no longer a pure number, but had a unit of measurement, the reciprocal mole. While it is rare to use units of amount of other than the mole, the Avogadro constant can also be expressed in units such as the pound mole. NA = 7026273159734000000♠2. 73159734×1026 −1 = 7025170724843400000♠1. 707248434×1025 −1 Avogadros constant is a factor between macroscopic and microscopic observations of nature. As such, it provides the relationship between other physical constants and properties. The Avogadro constant also enters into the definition of the atomic mass unit. The earliest accurate method to measure the value of the Avogadro constant was based on coulometry
16.
Mole (unit)
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The mole is the unit of measurement in the International System of Units for amount of substance. This number is expressed by the Avogadro constant, which has a value of 6. 022140857×1023 mol−1, the mole is one of the base units of the SI, and has the unit symbol mol. The mole is used in chemistry as a convenient way to express amounts of reactants and products of chemical reactions. For example, the chemical equation 2 H2 + O2 →2 H2O implies that 2 moles of dihydrogen and 1 mole of dioxygen react to form 2 moles of water. The mole may also be used to express the number of atoms, ions, the concentration of a solution is commonly expressed by its molarity, defined as the number of moles of the dissolved substance per litre of solution. For example, the relative molecular mass of natural water is about 18.015, therefore. The term gram-molecule was formerly used for essentially the same concept, the term gram-atom has been used for a related but distinct concept, namely a quantity of a substance that contains Avogadros number of atoms, whether isolated or combined in molecules. Thus, for example,1 mole of MgBr2 is 1 gram-molecule of MgBr2 but 3 gram-atoms of MgBr2, in honor of the unit, some chemists celebrate October 23, which is a reference to the 1023 scale of the Avogadro constant, as Mole Day. Some also do the same for February 6 and June 2, thus, by definition, one mole of pure 12C has a mass of exactly 12 g. It also follows from the definition that X moles of any substance will contain the number of molecules as X moles of any other substance. The mass per mole of a substance is called its molar mass, the number of elementary entities in a sample of a substance is technically called its amount. Therefore, the mole is a convenient unit for that physical quantity, one can determine the chemical amount of a known substance, in moles, by dividing the samples mass by the substances molar mass. Other methods include the use of the volume or the measurement of electric charge. The mass of one mole of a substance depends not only on its molecular formula, since the definition of the gram is not mathematically tied to that of the atomic mass unit, the number NA of molecules in a mole must be determined experimentally. The value adopted by CODATA in 2010 is NA =6. 02214129×1023 ±0. 00000027×1023, in 2011 the measurement was refined to 6. 02214078×1023 ±0. 00000018×1023. The number of moles of a sample is the sample mass divided by the mass of the material. The history of the mole is intertwined with that of mass, atomic mass unit, Avogadros number. The first table of atomic mass was published by John Dalton in 1805
17.
Astronomy
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Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
18.
Cosmology
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Cosmology is the study of the origin, evolution, and eventual fate of the universe. The term cosmology was first used in English in 1656 in Thomas Blounts Glossographia, religious or mythological cosmology is a body of beliefs based on mythological, religious, and esoteric literature and traditions of creation and eschatology. Physical cosmology is studied by scientists, such as astronomers and physicists, as well as philosophers, such as metaphysicians, philosophers of physics, and philosophers of space and time. Because of this scope with philosophy, theories in physical cosmology may include both scientific and non-scientific propositions, and may depend upon assumptions that can not be tested. Cosmology differs from astronomy in that the former is concerned with the Universe as a whole while the latter deals with individual celestial objects. Theoretical astrophysicist David N. Spergel has described cosmology as a science because when we look out in space. Physics and astrophysics have played a role in shaping the understanding of the universe through scientific observation. Physical cosmology was shaped through both mathematics and observation in an analysis of the whole universe, cosmogony studies the origin of the Universe, and cosmography maps the features of the Universe. In Diderots Encyclopédie, cosmology is broken down into uranology, aerology, geology, metaphysical cosmology has also been described as the placing of man in the universe in relationship to all other entities. Physical cosmology is the branch of physics and astrophysics that deals with the study of the physical origins and it also includes the study of the nature of the Universe on a large scale. In its earliest form, it was what is now known as celestial mechanics, greek philosophers Aristarchus of Samos, Aristotle, and Ptolemy proposed different cosmological theories. The geocentric Ptolemaic system was the theory until the 16th century when Nicolaus Copernicus. This is one of the most famous examples of epistemological rupture in physical cosmology, when Isaac Newton published the Principia Mathematica in 1687, he finally figured out how the heavens moved. A fundamental difference between Newtons cosmology and those preceding it was the Copernican principle—that the bodies on earth obey the same laws as all the celestial bodies. This was a crucial advance in physical cosmology. Physicists began changing the assumption that the Universe was static and unchanging, in 1922 Alexander Friedmann introduced the idea of an expanding universe that contained moving matter. In parallel to this approach to cosmology, one long-standing debate about the structure of the cosmos was coming to a climax. This difference of ideas came to a climax with the organization of the Great Debate on 26 April 1920 at the meeting of the U. S. National Academy of Sciences in Washington, D. C
19.
Big Bang model
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The Big Bang theory is the prevailing cosmological model for the universe from the earliest known periods through its subsequent large-scale evolution. If the known laws of physics are extrapolated to the highest density regime, detailed measurements of the expansion rate of the universe place this moment at approximately 13.8 billion years ago, which is thus considered the age of the universe. After the initial expansion, the universe cooled sufficiently to allow the formation of subatomic particles, giant clouds of these primordial elements later coalesced through gravity in halos of dark matter, eventually forming the stars and galaxies visible today. Since Georges Lemaître first noted in 1927 that a universe could be traced back in time to an originating single point. More recently, measurements of the redshifts of supernovae indicate that the expansion of the universe is accelerating, the known physical laws of nature can be used to calculate the characteristics of the universe in detail back in time to an initial state of extreme density and temperature. American astronomer Edwin Hubble observed that the distances to faraway galaxies were strongly correlated with their redshifts, assuming the Copernican principle, the only remaining interpretation is that all observable regions of the universe are receding from all others. Since we know that the distance between galaxies increases today, it must mean that in the past galaxies were closer together, the continuous expansion of the universe implies that the universe was denser and hotter in the past. Large particle accelerators can replicate the conditions that prevailed after the early moments of the universe, resulting in confirmation, however, these accelerators can only probe so far into high energy regimes. Consequently, the state of the universe in the earliest instants of the Big Bang expansion is still poorly understood, the first subatomic particles to be formed included protons, neutrons, and electrons. Though simple atomic nuclei formed within the first three minutes after the Big Bang, thousands of years passed before the first electrically neutral atoms formed, the majority of atoms produced by the Big Bang were hydrogen, along with helium and traces of lithium. Giant clouds of primordial elements later coalesced through gravity to form stars and galaxies. The framework for the Big Bang model relies on Albert Einsteins theory of relativity and on simplifying assumptions such as homogeneity. The governing equations were formulated by Alexander Friedmann, and similar solutions were worked on by Willem de Sitter, extrapolation of the expansion of the universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past. This singularity indicates that general relativity is not a description of the laws of physics in this regime. How closely models based on general relativity alone can be used to extrapolate toward the singularity is debated—certainly no closer than the end of the Planck epoch. This primordial singularity is itself called the Big Bang, but the term can also refer to a more generic early hot. The agreement of independent measurements of this age supports the model that describes in detail the characteristics of the universe. The earliest phases of the Big Bang are subject to much speculation, in the most common models the universe was filled homogeneously and isotropically with a very high energy density and huge temperatures and pressures and was very rapidly expanding and cooling
20.
Observable universe
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There are at least two trillion galaxies in the observable universe, containing more stars than all the grains of sand on planet Earth. Assuming the universe is isotropic, the distance to the edge of the universe is roughly the same in every direction. That is, the universe is a spherical volume centered on the observer. Every location in the Universe has its own universe, which may or may not overlap with the one centered on Earth. The word observable used in this sense does not depend on modern technology actually permits detection of radiation from an object in this region. It simply indicates that it is possible in principle for light or other signals from the object to reach an observer on Earth, in practice, we can see light only from as far back as the time of photon decoupling in the recombination epoch. That is when particles were first able to emit photons that were not quickly re-absorbed by other particles, before then, the Universe was filled with a plasma that was opaque to photons. The detection of gravitational waves indicates there is now a possibility of detecting signals from before the recombination epoch. The surface of last scattering is the collection of points in space at the distance that photons from the time of photon decoupling just reach us today. These are the photons we detect today as cosmic microwave background radiation, however, with future technology, it may be possible to observe the still older relic neutrino background, or even more distant events via gravitational waves. It is estimated that the diameter of the universe is about 28.5 gigaparsecs. The total mass of matter in the universe can be calculated using the critical density. Some parts of the Universe are too far away for the light emitted since the Big Bang to have had time to reach Earth. In the future, light from distant galaxies will have had time to travel. This fact can be used to define a type of cosmic event horizon whose distance from the Earth changes over time, both popular and professional research articles in cosmology often use the term universe to mean observable universe. It is plausible that the galaxies within our observable universe represent only a fraction of the galaxies in the Universe. If the Universe is finite but unbounded, it is possible that the Universe is smaller than the observable universe. In this case, what we take to be very distant galaxies may actually be duplicate images of nearby galaxies and it is difficult to test this hypothesis experimentally because different images of a galaxy would show different eras in its history, and consequently might appear quite different
21.
Ergodic hypothesis
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Liouvilles Theorem states that, for Hamiltonian systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble. Thus, if the microstates are uniformly distributed in phase space initially, but Liouvilles theorem does not imply that the ergodic hypothesis holds for all Hamiltonian systems. The ergodic hypothesis is often assumed in the analysis of computational physics. The analyst would assume that the average of a process parameter over time and this assumption that it is as good to simulate a system over a long time as it is to make many independent realizations of the same system is not always correct. Assumption of the ergodic hypothesis allows proof that certain types of perpetual motion machines of the second kind are impossible, the figure above displays situations where the ergodic hypothesis does and does not hold for a simplistic model of an ideal gas. If the walls are smooth and circular, the ergodic hypothesis does not hold. But, by Liouvilles theorem, if all regions of space are equally populated at time, t=0. No reflective trap or Maxwells demon will unmix a gas that has randomly filled both containers with equal density and pressure, the fact that macroscopic systems often violate the literal form of the ergodic hypothesis is an example of spontaneous symmetry breaking. Also conventional glasses violate ergodicity in a complicated manner, Ergodic theory is a branch of mathematics which deals with dynamical systems that satisfy a version of this hypothesis, phrased in the language of measure theory. Poincaré recurrence theorem Loschmidts paradox Ergodic theory, a branch of mathematics concerned with a general formulation of ergodicity Ergodic process Ergodicity
22.
Combinatorics
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Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general methods were developed. One of the oldest and most accessible parts of combinatorics is graph theory, Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist or a combinatorist, basic combinatorial concepts and enumerative results appeared throughout the ancient world. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, which was shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle, in the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for enumerative, graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, in part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis, in contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Partition theory studies various enumeration and asymptotic problems related to integer partitions, originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory, graphs are basic objects in combinatorics
23.
Factorial
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In mathematics, the factorial of a non-negative integer n, denoted by n. is the product of all positive integers less than or equal to n. =5 ×4 ×3 ×2 ×1 =120, the value of 0. is 1, according to the convention for an empty product. The factorial operation is encountered in areas of mathematics, notably in combinatorics, algebra. Its most basic occurrence is the fact there are n. ways to arrange n distinct objects into a sequence. This fact was known at least as early as the 12th century, fabian Stedman, in 1677, described factorials as applied to change ringing. After describing a recursive approach, Stedman gives a statement of a factorial, Now the nature of these methods is such, the factorial function is formally defined by the product n. = ∏ k =1 n k, or by the relation n. = {1 if n =0. The factorial function can also be defined by using the rule as n. All of the above definitions incorporate the instance 0, =1, in the first case by the convention that the product of no numbers at all is 1. This is convenient because, There is exactly one permutation of zero objects, = n. ×, valid for n >0, extends to n =0. It allows for the expression of many formulae, such as the function, as a power series. It makes many identities in combinatorics valid for all applicable sizes, the number of ways to choose 0 elements from the empty set is =0. More generally, the number of ways to choose n elements among a set of n is = n. n, the factorial function can also be defined for non-integer values using more advanced mathematics, detailed in the section below. This more generalized definition is used by advanced calculators and mathematical software such as Maple or Mathematica, although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics. There are n. different ways of arranging n distinct objects into a sequence, often factorials appear in the denominator of a formula to account for the fact that ordering is to be ignored. A classical example is counting k-combinations from a set with n elements, one can obtain such a combination by choosing a k-permutation, successively selecting and removing an element of the set, k times, for a total of n k _ = n ⋯ possibilities. This however produces the k-combinations in an order that one wishes to ignore, since each k-combination is obtained in k. different ways. This number is known as the coefficient, because it is also the coefficient of Xk in n
24.
Permutation
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These differ from combinations, which are selections of some members of a set where order is disregarded. For example, written as tuples, there are six permutations of the set, namely and these are all the possible orderings of this three element set. As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters, in this example, the letters are already ordered in the original word and the anagram is a reordering of the letters. The study of permutations of finite sets is a topic in the field of combinatorics, Permutations occur, in more or less prominent ways, in almost every area of mathematics. For similar reasons permutations arise in the study of sorting algorithms in computer science, the number of permutations of n distinct objects is n factorial, usually written as n. which means the product of all positive integers less than or equal to n. In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself and that is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f, the collection of such permutations form a group called the symmetric group of S. The key to this structure is the fact that the composition of two permutations results in another rearrangement. Permutations may act on structured objects by rearranging their components, or by certain replacements of symbols, in elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set, fabian Stedman in 1677 described factorials when explaining the number of permutations of bells in change ringing. Starting from two bells, first, two must be admitted to be varied in two ways which he illustrates by showing 12 and 21 and he then explains that with three bells there are three times two figures to be produced out of three which again is illustrated. His explanation involves cast away 3, and 1.2 will remain, cast away 2, and 1.3 will remain, cast away 1, and 2.3 will remain. He then moves on to four bells and repeats the casting away argument showing that there will be four different sets of three, effectively this is an recursive process. He continues with five bells using the casting method and tabulates the resulting 120 combinations. At this point he gives up and remarks, Now the nature of these methods is such, in modern mathematics there are many similar situations in which understanding a problem requires studying certain permutations related to it. There are two equivalent common ways of regarding permutations, sometimes called the active and passive forms, or in older terminology substitutions and permutations, which form is preferable depends on the type of questions being asked in a given discipline. The active way to regard permutations of a set S is to them as the bijections from S to itself. Thus, the permutations are thought of as functions which can be composed with each other, forming groups of permutations
25.
Stirling's approximation
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In mathematics, Stirlings approximation is an approximation for factorials. It is a good-quality approximation, leading to accurate results even for small values of n and it is named after James Stirling, though it was first stated by Abraham de Moivre. The formula as used in applications is ln n. = n ln n − n + O or, for instance in the lower bound for comparison sorting. N n +12 e − n is always between √2π =2.5066. and e =2.71828, the formula, together with precise estimates of its error, can be derived as follows. Instead of approximating n. one considers its natural logarithm as this is a slowly varying function, take limits to find that lim n → ∞ =1 − ∑ k =2 m k B k k + lim n → ∞ R m, n. = n ln +12 ln n + y + ∑ k =2 m k B k k n k −1 + O. Taking the exponential of both sides, and choosing any positive m, we get a formula involving an unknown quantity ey. For m =1, the formula is n, the quantity ey can be found by taking the limit on both sides as n tends to infinity and using Wallis product, which shows that ey = √2π. Therefore, we get Stirlings formula, n. =2 π n n, the formula may also be obtained by repeated integration by parts, and the leading term can be found through Laplaces method. Stirlings formula, without the factor √2πn that is irrelevant in applications. = ∑ j =1 n ln j with an integral, an alternative formula for n. using the gamma function is n. = ∫0 ∞ x n e − x d x. Rewriting and changing variables x = ny one gets n. = ∫0 ∞ e n ln x − x d x = e n ln n n ∫0 ∞ e n d y. Applying Laplaces method we have, ∫0 ∞ e n d y ∼2 π n e − n which recovers the Stirlings formula, ∼ e n ln n n 2 π n e − n =2 π n n. In fact further corrections can also be obtained using Laplaces method, for example, computing two-order expansion using Laplaces method yields ∫0 ∞ e n d y ∼2 π n e − n and gives Stirlings formula to two orders, n. ∼ e n ln n n 2 π n e − n =2 π n n, Stirlings formula is in fact the first approximation to the following series, n. ∼2 π n n. An explicit formula for the coefficients in this series was given by G. Nemes, the first graph in this section shows the relative error vs. n, for 1 through all 5 terms listed above. As n → ∞, the error in the series is asymptotically equal to the first omitted term
26.
Logarithm
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In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, in simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, the logarithm of x to base b, denoted logb, is the unique real number y such that by = x. For example, log2 =6, as 64 =26, the logarithm to base 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the e as its base, its use is widespread in mathematics and physics. The binary logarithm uses base 2 and is used in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations and they were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes, for example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios. In chemistry, pH is a measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and they describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant, it has uses in public-key cryptography, the idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power. For example, the power of 2 is 8, because 8 is the product of three factors of 2,23 =2 ×2 ×2 =8. It follows that the logarithm of 8 with respect to base 2 is 3, the third power of some number b is the product of three factors equal to b. More generally, raising b to the power, where n is a natural number, is done by multiplying n factors equal to b. The n-th power of b is written bn, so that b n = b × b × ⋯ × b ⏟ n factors, exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b−1 is the reciprocal of b, that is, the logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x
27.
Pathological (mathematics)
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In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive, the opposite is well-behaved. A classic example is the Weierstrass function, which is continuous everywhere, the sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable, so there are at least as many such functions as differentiable functions. In fact, by the Baire category theorem one can show that continuous functions are typically or generically nowhere differentiable, to quote Henri Poincaré, Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose, more of continuity, or less of continuity, more derivatives, and so forth. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions and it is the beginner that would have to be set grappling with this teratologic museum. In cases of pathology, often most or almost all examples of a phenomenon are pathological, for example, the set of rational numbers is countable, but the set of irrational numbers is uncountable, almost all real numbers are irrational, in these senses. In this case, pathologies are not the rare exceptions but the most common, mathematicians very frequently speak of whether a mathematical object—a function, a set, a space of one sort or another—is well-behaved. The term has no fixed formal definition, and is dependent on context, mathematical interests, fashion, to ensure that an object is well-behaved mathematicians introduce further axioms to narrow down the domain of study. This has the benefit of making analysis easier, but cuts down on the generality of any conclusions reached, concepts like non-Euclidean geometry were once considered ill-behaved, but are now common objects of study. In both pure and applied mathematics, well-behaved also means not violating any assumptions needed to apply whatever analysis is being discussed. The opposite case is usually labeled pathological and it is not unusual to have situations in which most cases are pathological, but the pathological cases will not arise in practice unless constructed deliberately. The term well-behaved is generally applied in an absolute sense—either something is well-behaved or it is not, for example, In algorithmic inference, a well-behaved statistic is monotonic, well-defined, and sufficient. A meromorphic function is a ratio of two well-behaved functions, in the sense of two functions being holomorphic. In probability, events contained in the probability spaces corresponding sigma-algebra are well-behaved, unusually, the term could also be applied in a comparative sense, In calculus, Analytic functions are better-behaved than general smooth functions. Smooth functions are better-behaved than general differentiable functions, continuous differentiable functions are better-behaved than general continuous functions. The larger the number of times the function can be differentiated, continuous functions are better-behaved than Riemann-integrable functions on compact sets. Riemann-integrable functions are better-behaved than Lebesgue-integrable functions, Lebesgue-integrable functions are better-behaved than general functions. In topology, continuous functions are better-behaved than discontinuous ones, euclidean space is better-behaved than non-Euclidean geometry
28.
Harvey Friedman
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Harvey Friedman is a mathematical logician at Ohio State University in Columbus, Ohio. He has worked on mathematics, a project intended to derive the axioms of mathematics from the theorems considered to be necessary. Friedman earned his Ph. D. from the Massachusetts Institute of Technology in 1967, Friedman received the Alan T. Waterman Award in 1984. He delivered the Tarski Lectures in 2007, in 1967, Friedman was listed in the Guinness Book of World Records for being the worlds youngest professor when he taught at Stanford University at age 18 as an assistant professor of philosophy. He has also been a professor of mathematics and a professor of music and he officially retired in July 2012. Friedman is the brother of mathematician Sy Friedman, friedmans grand conjecture L. A. Harrington et al. eds
29.
Seth Lloyd
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Seth Lloyd is a professor of mechanical engineering and physics at the Massachusetts Institute of Technology. He refers to himself as a quantum mechanic and his research area is the interplay of information with complex systems, especially quantum systems. Lloyd was born on August 2,1960 and he graduated from Phillips Academy in 1978 and received a bachelor of arts degree from Harvard College in 1982. He earned a certificate of advanced study in mathematics and a master of philosophy degree from Cambridge University in 1983 and 1984, from 1991 to 1994, he was a postdoctoral fellow at Los Alamos National Laboratory, where he worked at the Center for Nonlinear Systems on quantum computation. In 1994, he joined the faculty of the Department of Mechanical Engineering at MIT, since 1988, Lloyd has also been an external faculty member at the Santa Fe Institute. In his 2006 book, Programming the Universe, Lloyd contends that the universe itself is one big quantum computer producing what we see around us, and ourselves, as it runs a cosmic program. According to Lloyd, once we understand the laws of physics completely, Lloyd states that we could have the whole universe simulated in a computer in 600 years provided that computational power increases according to Moores Law. However, Lloyd shows that there are limits to rapid growth in a finite universe. Lloyd is principal investigator at the MIT Research Laboratory of Electronics and his most recent work has focused on the role of quantum phenomena such as coherence in biological phenomena, especially photosynthesis. He has also collaborated in work to exploit these phenomena technologically 2007 Fellow of the American Physical Society 2012 International Quantum Communication Award Lloyd, black Holes, Demons and the Loss of Coherence, How complex systems get information, and what they do with it. Lloyd, S. Programming the Universe, A Quantum Computer Scientist Takes On the Cosmos, Knopf, March 14,2006,240 p
30.
Holographic principle
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First proposed by Gerard t Hooft, it was given a precise string-theory interpretation by Leonard Susskind who combined his ideas with previous ones of t Hooft and Charles Thorn. As pointed out by Raphael Bousso, Thorn observed in 1978 that string theory admits a lower-dimensional description in which gravity emerges from it in what would now be called a holographic way. Cosmological holography has not been made mathematically precise, partly because the horizon has a non-zero area. The holographic principle was inspired by black hole thermodynamics, which conjectures that the entropy in any region scales with the radius squared. In the case of a hole, the insight was that the informational content of all the objects that have fallen into the hole might be entirely contained in surface fluctuations of the event horizon. The holographic principle resolves the black hole information paradox within the framework of string theory, however, there exist classical solutions to the Einstein equations that allow values of the entropy larger than those allowed by an area law, hence in principle larger than those of a black hole. These are the so-called Wheelers bags of gold, the existence of such solutions conflicts with the holographic interpretation, and their effects in a quantum theory of gravity including the holographic principle are not yet fully understood. An object with relatively high entropy is microscopically random, like a hot gas, a known configuration of classical fields has zero entropy, there is nothing random about electric and magnetic fields, or gravitational waves. Since black holes are exact solutions of Einsteins equations, they were not to have any entropy either. But Jacob Bekenstein noted that this leads to a violation of the law of thermodynamics. If one throws a hot gas with entropy into a hole, once it crosses the event horizon. The random properties of the gas would no longer be seen once the black hole had absorbed the gas and settled down. One way of salvaging the second law is if black holes are in random objects with an entropy that increases by an amount greater than the entropy of the consumed gas. Bekenstein assumed that black holes are maximum entropy objects—that they have more entropy than anything else in the same volume, in a sphere of radius R, the entropy in a relativistic gas increases as the energy increases. The only known limit is gravitational, when there is too much energy the gas collapses into a black hole, Bekenstein used this to put an upper bound on the entropy in a region of space, and the bound was proportional to the area of the region. He concluded that the black hole entropy is proportional to the area of the event horizon. Stephen Hawking had shown earlier that the horizon area of a collection of black holes always increases with time. The horizon is a boundary defined by light-like geodesics, it is those light rays that are just barely unable to escape, if neighboring geodesics start moving toward each other they eventually collide, at which point their extension is inside the black hole
31.
Big Bang
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The Big Bang theory is the prevailing cosmological model for the universe from the earliest known periods through its subsequent large-scale evolution. If the known laws of physics are extrapolated to the highest density regime, detailed measurements of the expansion rate of the universe place this moment at approximately 13.8 billion years ago, which is thus considered the age of the universe. After the initial expansion, the universe cooled sufficiently to allow the formation of subatomic particles, giant clouds of these primordial elements later coalesced through gravity in halos of dark matter, eventually forming the stars and galaxies visible today. Since Georges Lemaître first noted in 1927 that a universe could be traced back in time to an originating single point. More recently, measurements of the redshifts of supernovae indicate that the expansion of the universe is accelerating, the known physical laws of nature can be used to calculate the characteristics of the universe in detail back in time to an initial state of extreme density and temperature. American astronomer Edwin Hubble observed that the distances to faraway galaxies were strongly correlated with their redshifts, assuming the Copernican principle, the only remaining interpretation is that all observable regions of the universe are receding from all others. Since we know that the distance between galaxies increases today, it must mean that in the past galaxies were closer together, the continuous expansion of the universe implies that the universe was denser and hotter in the past. Large particle accelerators can replicate the conditions that prevailed after the early moments of the universe, resulting in confirmation, however, these accelerators can only probe so far into high energy regimes. Consequently, the state of the universe in the earliest instants of the Big Bang expansion is still poorly understood, the first subatomic particles to be formed included protons, neutrons, and electrons. Though simple atomic nuclei formed within the first three minutes after the Big Bang, thousands of years passed before the first electrically neutral atoms formed, the majority of atoms produced by the Big Bang were hydrogen, along with helium and traces of lithium. Giant clouds of primordial elements later coalesced through gravity to form stars and galaxies. The framework for the Big Bang model relies on Albert Einsteins theory of relativity and on simplifying assumptions such as homogeneity. The governing equations were formulated by Alexander Friedmann, and similar solutions were worked on by Willem de Sitter, extrapolation of the expansion of the universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past. This singularity indicates that general relativity is not a description of the laws of physics in this regime. How closely models based on general relativity alone can be used to extrapolate toward the singularity is debated—certainly no closer than the end of the Planck epoch. This primordial singularity is itself called the Big Bang, but the term can also refer to a more generic early hot. The agreement of independent measurements of this age supports the model that describes in detail the characteristics of the universe. The earliest phases of the Big Bang are subject to much speculation, in the most common models the universe was filled homogeneously and isotropically with a very high energy density and huge temperatures and pressures and was very rapidly expanding and cooling
32.
Exponential growth
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Exponential decay occurs in the same way when the growth rate is negative. In the case of a domain of definition with equal intervals, it is also called geometric growth or geometric decay. In either exponential growth or exponential decay, the ratio of the rate of change of the quantity to its current size remains constant over time. The formula for growth of a variable x at the growth rate r. This formula is transparent when the exponents are converted to multiplication, in this way, each increase in the exponent by a full interval can be seen to increase the previous total by another five percent. Since the time variable, which is the input to function, occurs as the exponent. Biology The number of microorganisms in a culture will increase exponentially until an essential nutrient is exhausted, typically the first organism splits into two daughter organisms, who then each split to form four, who split to form eight, and so on. Because exponential growth indicates constant growth rate, it is assumed that exponentially growing cells are at a steady-state. However, cells can grow exponentially at a constant rate while remodelling their metabolism, a virus typically will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people, human population, if the number of births and deaths per person per year were to remain at current levels. This means that the time of the American population is approximately 50 years. Physics Avalanche breakdown within a dielectric material, a free electron becomes sufficiently accelerated by an externally applied electrical field that it frees up additional electrons as it collides with atoms or molecules of the dielectric media. These secondary electrons also are accelerated, creating larger numbers of free electrons, the resulting exponential growth of electrons and ions may rapidly lead to complete dielectric breakdown of the material. Each uranium nucleus that undergoes fission produces multiple neutrons, each of which can be absorbed by adjacent uranium atoms, due to the exponential rate of increase, at any point in the chain reaction 99% of the energy will have been released in the last 4.6 generations. It is an approximation to think of the first 53 generations as a latency period leading up to the actual explosion. Economics Economic growth is expressed in terms, implying exponential growth. For example, U. S. GDP per capita has grown at a rate of approximately two percent since World War 2. Finance Compound interest at a constant interest rate provides exponential growth of the capital, pyramid schemes or Ponzi schemes also show this type of growth resulting in high profits for a few initial investors and losses among great numbers of investors
33.
P versus NP problem
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The P versus NP problem is a major unsolved problem in computer science. Informally speaking, it asks whether every problem whose solution can be verified by a computer can also be quickly solved by a computer. The underlying issues were first discussed in the 1950s, in letters from John Nash to the National Security Agency and it is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US$1,000,000 prize for the first correct solution. The general class of questions for which some algorithm can provide an answer in time is called class P or just P. For some questions, there is no way to find an answer quickly. The class of questions for which an answer can be verified in polynomial time is called NP, consider the subset sum problem, an example of a problem that is easy to verify, but whose answer may be difficult to compute. Given a set of integers, does some nonempty subset of them sum to 0, for instance, does a subset of the set add up to 0. The answer yes, because the subset adds up to zero can be verified with three additions. There is no algorithm to find such a subset in polynomial time. An answer to the P = NP question would determine whether problems that can be verified in polynomial time, like the subset-sum problem, can also be solved in polynomial time. Although the P versus NP problem was defined in 1971, there were previous inklings of the problems involved, the difficulty of proof. In 1955, mathematician John Nash wrote a letter to the NSA, if proved this would imply what we today would call P ≠ NP, since a proposed key can easily be verified in polynomial time. Another mention of the problem occurred in a 1956 letter written by Kurt Gödel to John von Neumann. The most common resources are time and space, in such analysis, a model of the computer for which time must be analyzed is required. Typically such models assume that the computer is deterministic and sequential, arguably the biggest open question in theoretical computer science concerns the relationship between those two classes, Is P equal to NP. In 2012,10 years later, the poll was repeated. To attack the P = NP question, the concept of NP-completeness is very useful, NP-complete problems are a set of problems to each of which any other NP-problem can be reduced in polynomial time, and whose solution may still be verified in polynomial time. That is, any NP problem can be transformed into any of the NP-complete problems, informally, an NP-complete problem is an NP problem that is at least as tough as any other problem in NP
. Bibcode:2002PhRvL..88w7901L. doi:10.1103/PhysRevLett.88.237901. PMID 12059399.
