In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension exceeds the topological dimension. Fractals tend to appear nearly the same at different levels, as is illustrated here in the successively small magnifications of the Mandelbrot set. Fractals exhibit similar patterns at small scales called self similarity known as expanding symmetry or unfolding symmetry. One way that fractals are different from finite geometric figures is the way. Doubling the edge lengths of a polygon multiplies its area by four, two raised to the power of two. If the radius of a sphere is doubled, its volume scales by eight, two to the power of three. However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power, not an integer; this power is called the fractal dimension of the fractal, it exceeds the fractal's topological dimension. Analytically, fractals are nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still 1-dimensional, its fractal dimension indicates that it resembles a surface.
Starting in the 17th century with notions of recursion, fractals have moved through rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, Karl Weierstrass, on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus, meaning "broken" or "fractured", used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature. There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard useful. That's fractals." More formally, in 1982 Mandelbrot stated that "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension exceeds the topological dimension."
Seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way." Still Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants". The consensus is that theoretical fractals are infinitely self-similar and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images and sounds and found in nature, art and law. Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals; the word "fractal" has different connotations for laymen as opposed to mathematicians, where the layman is more to be familiar with fractal art than the mathematical concept.
The mathematical concept is difficult to define formally for mathematicians, but key features can be understood with little mathematical background. The feature of "self-similarity", for instance, is understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer invisible, new structure. If this is done on fractals, however, no new detail appears. Self-similarity itself is not counter-intuitive; the difference for fractals is. This idea of being detailed relates to another feature that can be understood without mathematical background: Having a fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are perceived. A regular line, for instance, is conventionally understood to be one-dimensional. A solid square is understood to be two-dimensional. We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/r, there are a total of rn pieces.
Now, consider the Koch curve. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So by analogy, we can consider the "dimension" of the Koch curve as being the unique real number D that satisfies 3D = 4, which by no means is an integer! This number is; the fact th
Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic; the most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this defines an infinite number of instances, it is done in such a way that no loop or infinite chain of references can occur. In mathematics and computer science, a class of objects or methods exhibit recursive behavior when they can be defined by two properties: A simple base case —a terminating scenario that does not use recursion to produce an answer A set of rules that reduce all other cases toward the base caseFor example, the following is a recursive definition of a person's ancestors: One's parents are one's ancestors; the ancestors of one's ancestors are one's ancestors. The Fibonacci sequence is a classic example of recursion: Fib = 0 as base case 1, Fib = 1 as base case 2, For all integers n > 1, Fib:= Fib + Fib.
Many mathematical axioms are based upon recursive rules. For example, the formal definition of the natural numbers by the Peano axioms can be described as: 0 is a natural number, each natural number has a successor, a natural number. By this base case and recursive rule, one can generate the set of all natural numbers. Recursively defined mathematical objects include functions and fractals. There are various more tongue-in-cheek "definitions" of recursion. Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be'recursive'. To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules; the running of a procedure involves following the rules and performing the steps. An analogy: a procedure is like a written recipe. Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure.
For instance, a recipe might refer to cooking vegetables, another procedure that in turn requires heating water, so forth. However, a recursive procedure is where one of its steps calls for a new instance of the same procedure, like a sourdough recipe calling for some dough left over from the last time the same recipe was made; this creates the possibility of an endless loop. If properly defined, a recursive procedure is not easy for humans to perform, as it requires distinguishing the new from the old invocation of the procedure. For this reason recursive definitions are rare in everyday situations. An example could be the following procedure to find a way through a maze. Proceed forward until reaching either an exit or a branching point. If the point reached is an exit, terminate. Otherwise try each branch in turn, using the procedure recursively. Whether this defines a terminating procedure depends on the nature of the maze: it must not allow loops. In any case, executing the procedure requires recording all explored branching points, which of their branches have been exhaustively tried.
Linguist Noam Chomsky among many others has argued that the lack of an upper bound on the number of grammatical sentences in a language, the lack of an upper bound on grammatical sentence length, can be explained as the consequence of recursion in natural language. This can be understood in terms of a recursive definition of a syntactic category, such as a sentence. A sentence can have a structure in which what follows the verb is another sentence: Dorothy thinks witches are dangerous, in which the sentence witches are dangerous occurs in the larger one. So a sentence can be defined recursively as something with a structure that includes a noun phrase, a verb, optionally another sentence; this is just a special case of the mathematical definition of recursion. This provides a way of understanding the creativity of language—the unbounded number of grammatical sentences—because it predicts that sentences can be of arbitrary length: Dorothy thinks that Toto suspects that Tin Man said that.... There are many structures apart from sentences that can be defined recursively, therefore many ways in which a sentence can embed instances of one
Graphene is an allotrope of carbon consisting of a single layer of carbon atoms arranged in a hexagonal lattice. Graphene can be considered as an indefinitely large aromatic molecule, the ultimate case of the family of flat polycyclic aromatic hydrocarbons. Graphite, the most common allotrope of carbon, is a stack of graphene layers held together with weak bonds. Fullerenes and carbon nanotubes, two other forms of carbon, have structures similar to that of graphene. Graphene has many uncommon properties, it is the strongest material tested, conducts heat and electricity efficiently, is nearly transparent, yet opaque for a 1-atom-thick layer. Graphene shows a large and nonlinear diamagnetism, greater than that of graphite, can be levitated by neodymium magnets, it is a semimetal with small overlap between the conduction bands. Scientists theorized about graphene for years, it had been produced unintentionally in small quantities for centuries through the use of pencils and other similar graphite applications.
It was observed in electron microscopes in 1962, but it was studied only while supported on metal surfaces. The material was rediscovered and characterized in 2004 by Andre Geim and Konstantin Novoselov at the University of Manchester. Research was informed by existing theoretical descriptions of its composition and properties; this work resulted in the two winning the Nobel Prize in Physics in 2010 "for groundbreaking experiments regarding the two-dimensional material graphene." The name "graphene" is a combination of "graphite" and the suffix -ene, named by Hanns-Peter Boehm and colleagues, who produced and observed single-layer carbon foils in 1962. Boehm et al. introduced the term graphene in 1986 to describe single sheets of graphite. Graphene can be considered an "infinite alternant" polycyclic aromatic hydrocarbon; the International Union of Pure and Applied Chemistry notes, "previously, descriptions such as graphite layers, carbon layers, or carbon sheets have been used for the term graphene...it is incorrect to use for a single layer a term which includes the term graphite, which would imply a three-dimensional structure.
The term graphene should be used only when the reactions, structural relations or other properties of individual layers are discussed."Geim defined "isolated or free-standing graphene" as "a single atomic plane of graphite, which – and this is essential – is sufficiently isolated from its environment to be considered free-standing." This definition is narrower than the IUPAC definition and refers to cloven and suspended graphene. Other forms such as graphene grown on various metals, can become free-standing if, for example, suspended or transferred to silicon dioxide or silicon carbide. Graphene is a crystalline allotrope of carbon with 2-dimensional properties, its carbon atoms are packed densely in a regular atomic-scale chicken wire pattern. Each atom has four bonds: one σ bond with each of its three neighbors and one π-bond, oriented out of plane; the atoms are about 1.42 Å apart. Graphene's hexagonal lattice can be regarded as two interleaving triangular lattices; this perspective was used to calculate the band structure for a single graphite layer using a tight-binding approximation.
Graphene's stability is due to its packed carbon atoms and a sp2 orbital hybridization – a combination of orbitals s, px and py that constitute the σ-bond. The final pz electron makes up the π-bond; the π-bonds hybridize together to form the π ∗ - bands. These bands are responsible for most of graphene's notable electronic properties, via the half-filled band that permits free-moving electrons. Graphene sheets in solid form show evidence in diffraction for graphite's layering; this is true of some single-walled nanostructures. However, unlayered graphene with only rings has been found in the core of presolar graphite onions. TEM studies show faceting at defects in flat graphene sheets and suggest a role for two-dimensional crystallization from a melt. Graphene can self-repair holes in its sheets when exposed to molecules containing carbon, such as hydrocarbons. Bombarded with pure carbon atoms, the atoms align into hexagons filling the holes; the atomic structure of isolated, single-layer graphene is studied by TEM on sheets of graphene suspended between bars of a metallic grid.
Electron diffraction patterns showed the expected honeycomb lattice. Suspended graphene showed "rippling" of the flat sheet, with amplitude of about one nanometer; these ripples may be intrinsic to the material as a result of the instability of two-dimensional crystals, or may originate from the ubiquitous dirt seen in all TEM images of graphene. Atomic resolution real-space images of isolated, single-layer graphene on SiO2 substrates are available via scanning tunneling microscopy. Photoresist residue, which must be removed to obtain atomic-resolution images, may be the "adsorbates" observed in TEM images, may explain the observed rippling. Rippling on SiO2 is caused by conformation of graphene to the underlying SiO2 and is not intrinsic. Ab initio calculations show that a graphene sheet is thermodynamically unstable if its size is less than about 20 nm and becomes the most stable fullerene only for molecules larger than 24,000 atoms. Analogs are two-dimensional systems. Analogs can be systems in which the physics is easier to manipulate.
In those systems
A Bloch wave, named after Swiss physicist Felix Bloch, is a type of wavefunction for a particle in a periodically-repeating environment, most an electron in a crystal. A wavefunction ψ is a Bloch wave if it has the form: ψ = e i k ⋅ r u where r is position, ψ is the Bloch wave, u is a periodic function with the same periodicity as the crystal, k is a vector of real numbers called the crystal wave vector, e is Euler's number, i is the imaginary unit. In other words, if we multiply a plane wave by a periodic function, we get a Bloch wave. Bloch waves are important because of Bloch's theorem, which states that the energy eigenstates for an electron in a crystal can be written as Bloch waves; this fact underlies the concept of electronic band structures. These Bloch wave energy eigenstates are written with subscripts as ψn k, where n is a discrete index, called the band index, present because there are many different Bloch waves with the same k. Within a band, ψn k varies continuously with k. For any reciprocal lattice vector K, ψn k = ψn.
Therefore, all distinct Bloch waves occur for k-values within the first Brillouin zone of the reciprocal lattice. The most common example of Bloch's theorem is describing electrons in a crystal. However, a Bloch-wave description applies more to any wave-like phenomenon in a periodic medium. For example, a periodic dielectric in electromagnetism leads to photonic crystals, a periodic acoustic medium leads to phononic crystals, it is treated in the various forms of the dynamical theory of diffraction. Suppose an electron is in a Bloch state ψ = e i k ⋅ r u, where u is periodic with the same periodicity as the crystal lattice; the actual quantum state of the electron is determined by ψ, not k or u directly. This is important because u are not unique. If ψ can be written as above using k, it can be written using, where K is any reciprocal lattice vector. Therefore, wave vectors that differ by a reciprocal lattice vector are equivalent, in the sense that they characterize the same set of Bloch states.
The first Brillouin zone is a restricted set of values of k with the property that no two of them are equivalent, yet every possible k is equivalent to one vector in the first Brillouin zone. Therefore, if we restrict k to the first Brillouin zone every Bloch state has a unique k. Therefore, the first Brillouin zone is used to depict all of the Bloch states without redundancy, for example in a band structure, it is used for the same reason in many calculations; when k is multiplied by the reduced Planck's constant, it equals the electron's crystal momentum. Related to this, the group velocity of an electron can be calculated based on how the energy of a Bloch state varies with k. For a detailed example in which the consequences of Bloch's theorem are worked out in a specific situation, see the article: Particle in a one-dimensional lattice. Next, we prove Bloch's theorem: For electrons in a perfect crystal, there is a basis of wavefunctions with the properties: Each of these wavefunctions is an energy eigenstate Each of these wavefunctions is a Bloch wave, meaning that this wavefunction ψ can be written in the form ψ = e i k ⋅ r u where u has the same periodicity as the atomic structure of the crystal.
The defining property of a crystal is translational symmetry, which means that if the crystal is shifted an appropriate amount, it winds up with all its atoms in the same places. A three-dimensional crystal has three primitive lattice vectors a1, a2, a3. If the crystal is shifted by any of these three vectors, or a combination of them of the form n 1 a 1 + n 2 a 2 + n 3 a 3 where ni are three integers the atoms end up in the same set of locations as they started. Another helpful ingredient in the proof is the reciprocal lattice vectors; these are three vectors b1, b2, b3, with the property that ai · bi = 2π, but ai · bj = 0 when i ≠ j. Let T ^ n 1, n 2, n 3 denote a translation operator that shifts every
International Standard Serial Number
An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication, such as a magazine. The ISSN is helpful in distinguishing between serials with the same title. ISSN are used in ordering, interlibrary loans, other practices in connection with serial literature; the ISSN system was first drafted as an International Organization for Standardization international standard in 1971 and published as ISO 3297 in 1975. ISO subcommittee TC 46/SC 9 is responsible for maintaining the standard; when a serial with the same content is published in more than one media type, a different ISSN is assigned to each media type. For example, many serials are published both in electronic media; the ISSN system refers to these types as electronic ISSN, respectively. Conversely, as defined in ISO 3297:2007, every serial in the ISSN system is assigned a linking ISSN the same as the ISSN assigned to the serial in its first published medium, which links together all ISSNs assigned to the serial in every medium.
The format of the ISSN is an eight digit code, divided by a hyphen into two four-digit numbers. As an integer number, it can be represented by the first seven digits; the last code digit, which may be 0-9 or an X, is a check digit. Formally, the general form of the ISSN code can be expressed as follows: NNNN-NNNC where N is in the set, a digit character, C is in; the ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the check digit, C=5. To calculate the check digit, the following algorithm may be used: Calculate the sum of the first seven digits of the ISSN multiplied by its position in the number, counting from the right—that is, 8, 7, 6, 5, 4, 3, 2, respectively: 0 ⋅ 8 + 3 ⋅ 7 + 7 ⋅ 6 + 8 ⋅ 5 + 5 ⋅ 4 + 9 ⋅ 3 + 5 ⋅ 2 = 0 + 21 + 42 + 40 + 20 + 27 + 10 = 160 The modulus 11 of this sum is calculated. For calculations, an upper case X in the check digit position indicates a check digit of 10. To confirm the check digit, calculate the sum of all eight digits of the ISSN multiplied by its position in the number, counting from the right.
The modulus 11 of the sum must be 0. There is an online ISSN checker. ISSN codes are assigned by a network of ISSN National Centres located at national libraries and coordinated by the ISSN International Centre based in Paris; the International Centre is an intergovernmental organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, the ISDS Register otherwise known as the ISSN Register. At the end of 2016, the ISSN Register contained records for 1,943,572 items. ISSN and ISBN codes are similar in concept. An ISBN might be assigned for particular issues of a serial, in addition to the ISSN code for the serial as a whole. An ISSN, unlike the ISBN code, is an anonymous identifier associated with a serial title, containing no information as to the publisher or its location. For this reason a new ISSN is assigned to a serial each time it undergoes a major title change. Since the ISSN applies to an entire serial a new identifier, the Serial Item and Contribution Identifier, was built on top of it to allow references to specific volumes, articles, or other identifiable components.
Separate ISSNs are needed for serials in different media. Thus, the print and electronic media versions of a serial need separate ISSNs. A CD-ROM version and a web version of a serial require different ISSNs since two different media are involved. However, the same ISSN can be used for different file formats of the same online serial; this "media-oriented identification" of serials made sense in the 1970s. In the 1990s and onward, with personal computers, better screens, the Web, it makes sense to consider only content, independent of media; this "content-oriented identification" of serials was a repressed demand during a decade, but no ISSN update or initiative occurred. A natural extension for ISSN, the unique-identification of the articles in the serials, was the main demand application. An alternative serials' contents model arrived with the indecs Content Model and its application, the digital object identifier, as ISSN-independent initiative, consolidated in the 2000s. Only in 2007, ISSN-L was defined in the
Physical Review B
Physical Review B: Condensed Matter and Materials Physics is a peer-reviewed, scientific journal, published by the American Physical Society. The Editor of PRB is Laurens W. Molenkamp, it is part of the Physical Review family of journals. The current Editor in Chief is Michael Thoennessen. PRB publishes over 4500 papers a year, making it one of the largest physics journals in the world. According to the Journal Citation Reports, PRB's most recent impact factors have been 3.736 for 2014, 3.718 for 2015 and 3.836 for 2016. The focus of this journal is on new results in condensed matter physics, which includes a wide variety of subject areas, such as semiconductors, superconductivity, structure, phase transitions, nonordered systems, quantum solids, electronic structure, photonic crystals, mesoscopic systems, clusters, graphene, etc. PRB was created in 1970 by the split of the original Physical Review into four parts, based on subject matter. Peter D. Adams was the Editor from inception until 2012.
Anthony M. Begley is the Managing Editor. PRB has a reputation among professional physicists for publishing useful, comprehensive long papers in physics, it contains short papers in its Rapid Communications section, designed for research important enough to deserve special handling and speedy publication. The journal can be searched free via PROLA. Titles and abstracts can be viewed free but a journal subscription is needed to read the full text of papers. PRB and the other APS journals are available free at many US public libraries. PRB is rare among physics journals in that it has a staff of 12 full-time professional editors and does not employ the more common model of using part-time editors who are active researchers; the journal is available in print format but the archival version is the online one. Authors can pay extra charges to make their papers open access; such papers are published under the terms of the Creative Commons Attribution 3.0 License, the most permissive of the CC licenses, which permits authors and others to copy, distribute and adapt the work, provided that proper credit is given.
A small percentage of the PRB papers published are chosen by the PRB editors to be Editors' Suggestions, as seen at http://prb.aps.org. Artistic images from papers in the journal are published as a feature named "Kaleidoscope" at http://prb.aps.org/kaleidoscope. Physical Review B is indexed in the following bibliographic databases: American Physical Society#APS journals PRB home page