While the team's findings are not groundbreaking for number theory (as most of the relevant mathematics has been seen before in other forms), they may prove useful in a new research area known as "aperiodic order"—the study of non-repeating patterns.
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Schrödinger didn't know where the information is kept or how it is encoded, but his intuition that it is written into what he called an "aperiodic crystal" inspired Francis Crick, himself trained as a physicist, and James Watson when in 1953 they figured out how genetic information can be encoded in the molecular structure of the DNA molecule.
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In mathematics, an aperiodic semigroup is a semigroup S such that every element x ∈ S is aperiodic, that is, for each x there exists a positive integer n such that xn = xn + 1. An aperiodic monoid is an aperiodic semigroup which is a monoid.
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An aperiodic finite-state automaton (also called a counter-free automaton) is a finite-state automaton whose transition monoid is aperiodic.
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A set of prototiles is said to be aperiodic if every tiling with those prototiles is an aperiodic tiling. It is unknown whether there exists a single two- dimensional shape (called an einstein). that forms the prototile of an aperiodic tiling, but not of any periodic tiling. That is, the existence of a single-tile (monohedral) aperiodic prototile set is an open problem.
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An aperiodic sequence has at least n + 1 distinct factors of length n. A sequence is Sturmian if and only if it is balanced and aperiodic.
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The Penrose tiling is an example of an aperiodic tiling; every tiling it can produce lacks translational symmetry. An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non-periodic tilings. The Penrose tilings are the best-known examples of aperiodic tilings.
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A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic. The tilings obtained from an aperiodic set of tiles are often called aperiodic tilings, though strictly speaking it is the tiles themselves that are aperiodic. (The tiling itself is said to be "nonperiodic".) The first table explains the abbreviations used in the second table.
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Quasicrystal structures of Cd-Te appear to consist of atomic layers in which the atoms are arranged in a planar aperiodic pattern. Sometimes an energetical minimum or a maximum of entropy occur for such aperiodic structures. Steinhardt has shown that Gummelt's overlapping decagons allow the application of an extremal principle and thus provide the link between the mathematics of aperiodic tiling and the structure of quasicrystals. Faraday waves have been observed to form large patches of aperiodic patterns.
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Infinitely many distinct tilings may be obtained from a single aperiodic set of tiles.A set of aperiodic prototiles can always form uncountably many different tilings, even up to isometry, as proven by Nikolaï Dolbilin in his 1995 paper The Countability of a Tiling Family and the Periodicity of a Tiling The best-known examples of an aperiodic set of tiles are the various Penrose tiles. The known aperiodic sets of prototiles are seen on the list of aperiodic sets of tiles. The underlying undecidability of the domino problem implies that there exists no systematic procedure for deciding whether a given set of tiles can tile the plane.
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A task can be classified as either a periodic or aperiodic process.
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He discovered several new aperiodic tilings, each among the simplest known examples of aperiodic sets of tiles. He also showed how to generate tilings using lines in the plane as guides for lines marked on the tiles, now called "Ammann bars". The discovery of quasicrystals in 1982 changed the status of aperiodic tilings and Ammann's work from mere recreational mathematics to respectable academic research.
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That is, each tile in the tiling must be congruent to one of these prototiles. A tiling that has no periods is non-periodic. A set of prototiles is said to be aperiodic if all of its tilings are non-periodic, and in this case its tilings are also called aperiodic tilings. Penrose tilings are among the simplest known examples of aperiodic tilings of the plane by finite sets of prototiles.
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In particular, the celebrated Penrose tiling is an example of an aperiodic substitution tiling.
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In a strongly connected graph, if one defines a Markov chain on the vertices, in which the probability of transitioning from v to w is nonzero if and only if there is an edge from v to w, then this chain is aperiodic if and only if the graph is aperiodic. A Markov chain in which all states are recurrent has a strongly connected state transition graph, and the Markov chain is aperiodic if and only if this graph is aperiodic. Thus, aperiodicity of graphs is a useful concept in analyzing the aperiodicity of Markov chains. Aperiodicity is also an important necessary condition for solving the road coloring problem.
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Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood. Several methods for constructing aperiodic tilings are known.
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Grünbaum and Shephard, section 11.1. An even smaller set of six aperiodic tiles (based on Wang tiles) was discovered by Raphael M. Robinson in 1971. Roger Penrose discovered three more sets in 1973 and 1974, reducing the number of tiles needed to two, and Robert Ammann discovered several new sets in 1977. The aperiodic Penrose tilings can be generated not only by an aperiodic set of prototiles, but also by a substitution and by a cut-and-project method.
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There are a few constructions of aperiodic tilings known. Some constructions are based on infinite families of aperiodic sets of tiles. Those constructions which have been found are mostly constructed in a few ways, primarily by forcing some sort of non-periodic hierarchical structure. Despite this, the undecidability of the Domino Problem ensures that there must be infinitely many distinct principles of construction, and that in fact, there exist aperiodic sets of tiles for which there can be no proof of their aperiodicity.
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A finite semigroup is aperiodic if and only if it contains no nontrivial subgroups, so a synonym used (only?) in such contexts is group-free semigroup. In terms of Green's relations, a finite semigroup is aperiodic if and only if its H-relation is trivial. These two characterizations extend to group-bound semigroups. A celebrated result of algebraic automata theory due to Marcel-Paul Schützenberger asserts that a language is star-free if and only if its syntactic monoid is finite and aperiodic.
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After the discovery of quasicrystals aperiodic tilings become studied intensively by physicists and mathematicians. The cut-and-project method of N.G. de Bruijn for Penrose tilings eventually turned out to be an instance of the theory of Meyer sets. Today there is a large amount of literature on aperiodic tilings.
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The term aperiodic has been used in a wide variety of ways in the mathematical literature on tilings (and in other mathematical fields as well, such as dynamical systems or graph theory, with altogether different meanings). With respect to tilings the term aperiodic was sometimes used synonymously with the term non-periodic. A non-periodic tiling is simply one that is not fixed by any non-trivial translation. Sometimes the term described – implicitly or explicitly – a tiling generated by an aperiodic set of prototiles.
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The Socolar-Taylor tile forms two-dimensional aperiodic tilings, but is defined by combinatorial matching conditions rather than purely by its shape. In higher dimensions, the problem is solved: the Schmitt-Conway-Danzer tile is the prototile of a monohedral aperiodic tiling of three-dimensional Euclidean space, and cannot tile space periodically.
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There are few constructions of aperiodic tilings known, even forty years after Berger's groundbreaking construction. Some constructions are of infinite families of aperiodic sets of tiles. Those constructions which have been found are mostly constructed in a few ways, primarily by forcing some sort of non-periodic hierarchical structure. Despite this, the undecidability of the Domino Problem ensures that there must be infinitely many distinct principles of construction, and that in fact, there exist aperiodic sets of tiles for which there can be no proof of their aperiodicity.
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In recent years, aperiodic entrainment has been identified as an alternative form of entrainment that is of interest in biological rhythms.
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Conversely, if pu(n) ≤ n for some n, then u is ultimately periodic.Allouche & Shallit (2003) p.302 An aperiodic sequence is one which is not ultimately periodic. An aperiodic sequence has strictly increasing complexity function (this is the Morse–Hedlund theorem),Cassaigne & Nicolas (2010) p.166 so p(n) is at least n+1.Lothaire (2011) p.
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For such a coloring to exist at all, it is necessary that G be aperiodic.. The road coloring theorem states that aperiodicity is also sufficient for such a coloring to exist. Therefore, the road coloring problem can be stated briefly as: :Every finite strongly connected directed aperiodic graph of uniform out-degree has a synchronizing coloring.
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The physics of this discovery has revived the interest in incommensurate structures and frequencies suggesting to link aperiodic tilings with interference phenomena.
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The Schmitt–Conway–Danzer tile, a convex polyhedron that tiles space, is not a stereohedron because all of its tilings are aperiodic.
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However, random, aperiodic patterns may also be generated using carefully defined guiding patterns.L-W. Chang et al.IEDM 2010 Technical Digest, 752-755 (2010).
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Robert Berger (born 1938) is an applied mathematician, known for inventing the first aperiodic tiling using a set of 20,426 distinct tile shapes.
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An aperiodic graph. The cycles in this graph have lengths 5 and 6; therefore, there is no k > 1 that divides all cycle lengths. strongly connected graph with period three. In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer k > 1 that divides the length of every cycle of the graph.
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The second table contains all known aperiodic sets of tiles and gives some additional basic information about each set. This list of tiles is still incomplete.
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This idea — of finding sets of tiles that can only admit hierarchical structures — has been used in the construction of most known aperiodic sets of tiles to date.
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Combining Berger's undecidability result with Wang's observation shows that there must exist a finite set of Wang tiles that tiles the plane, but only aperiodically. This is similar to a Penrose tiling, or the arrangement of atoms in a quasicrystal. Although Berger's original set contained 20,426 tiles, he conjectured that smaller sets would work, including subsets of his set, and in his unpublished Ph.D. thesis, he reduced the number of tiles to 104. In later years, increasingly smaller sets were found... (Showed an aperiodic set of 13 tiles with 5 colors)... (Showed an aperiodic set of 11 tiles with 4 colors)} For example, a set of 13 aperiodic tiles was published by Karel Culik II in 1996.
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A state i is said to be ergodic if it is aperiodic and positive recurrent. In other words, a state i is ergodic if it is recurrent, has a period of 1, and has finite mean recurrence time. If all states in an irreducible Markov chain are ergodic, then the chain is said to be ergodic. It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state.
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The chair substitution tiling system. However, the tiles shown below force the chair substitution structure to emerge, and so are themselves aperiodic. Trilobite and Cross tiles enforce the chair substitution structure—they can only admit tilings in which the chair substitution can be discerned and so are aperiodic. The Penrose tiles, and shortly thereafter Amman's several different sets of tiles, were the first example based on explicitly forcing a substitution tiling structure to emerge.
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"However, as will be explained momentarily, differently colored pentagons will be considered to be different types of tiles." ; , shows the edge modifications needed to yield an aperiodic set of prototiles.
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Daniel Wise obtained his Ph.D. from Princeton University in 1996 supervised by Martin Bridson His thesis was titled non-positively curved squared complexes, aperiodic tilings, and non- residually finite groups.
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Charles Lewis Radin is an American mathematician, known for his work on aperiodic tilings and in particular for defining the pinwheel tiling and (with John Horton Conway) the quaquaversal tiling..
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If "a" and "b" represent two different materials or atomic bond lengths, the structure corresponding to a Fibonacci string is a Fibonacci quasicrystal, an aperiodic quasicrystal structure with unusual spectral properties.
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Only a few different kinds of constructions have been found. Notably, Jarkko Kari gave an aperiodic set of Wang tiles based on multiplications by 2 or 2/3 of real numbers encoded by lines of tiles (the encoding is related to Sturmian sequences made as the differences of consecutive elements of Beatty sequences), with the aperiodicity mainly relying on the fact that 2n/3m is never equal to 1 for any positive integers n and m. This method was later adapted by Goodman-Strauss to give a strongly aperiodic set of tiles in the hyperbolic plane. Shahar Mozes has found many alternative constructions of aperiodic sets of tiles, some in more exotic settings; for example in semi-simple Lie Groups.
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In any directed bipartite graph, all cycles have a length that is divisible by two. Therefore, no directed bipartite graph can be aperiodic. In any directed acyclic graph, it is a vacuous truth that every k divides all cycles (because there are no directed cycles to divide) so no directed acyclic graph can be aperiodic. And in any directed cycle graph, there is only one cycle, so every cycle's length is divisible by n, the length of that cycle.
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The sequence of tilings where b is centred at 1,2,4, \ldots,2^n,\ldots converges – in the local topology – to the periodic tiling consisting of as only. Thus T is not an aperiodic tiling, since its hull contains the periodic tiling For well-behaved tilings (e.g. substitution tilings with finitely many local patterns) holds: if a tiling is non-periodic and repetitive (i.e. each patch occurs in a uniformly dense way throughout the tiling), then it is aperiodic.
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Equivalently, a graph is aperiodic if the greatest common divisor of the lengths of its cycles is one; this greatest common divisor for a graph G is called the period of G.
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According to the solution of this problem , a strongly connected directed graph in which all vertices have the same outdegree has a synchronizable edge coloring if and only if it is aperiodic.
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An aperiodic sequence generated from tilings by two squares whose side lengths form the golden ratio Although the Pythagorean tiling is itself periodic (it has a square lattice of translational symmetries) its cross sections can be used to generate one-dimensional aperiodic sequences.. In the "Klotz construction" for aperiodic sequences (Klotz is a German word for a block), one forms a Pythagorean tiling with two squares whose sizes are chosen to make the ratio between the two side lengths be an irrational number x. Then, one chooses a line parallel to the sides of the squares, and forms a sequence of binary values from the sizes of the squares crossed by the line: a 0 corresponds to a crossing of a large square and a 1 corresponds to a crossing of a small square. In this sequence, the relative proportion of 0s and 1s will be in the ratio x:1. This proportion cannot be achieved by a periodic sequence of 0s and 1s, because it is irrational, so the sequence is aperiodic.
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Fibonacci based constructions are currently used to model physical systems with aperiodic order such as quasicrystals. Crystal growth techniques have been used to grow Fibonacci layered crystals and study their light scattering properties.
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A group C of individuals is said to reach a consensus if p_i(\infty)= p_j(\infty) for any i, j \in C . This means that, as a result of the learning process, in the limit they have the same belief on the subject. With a strongly connected and aperiodic network the whole group reaches a consensus. In general, any strongly connected and closed group C of individuals reaches a consensus for every initial vector of beliefs if and only if it is aperiodic.
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In 1973 and 1974, Roger Penrose discovered a family of aperiodic tilings, now called Penrose tilings. The first description was given in terms of 'matching rules' treating the prototiles as jigsaw puzzle pieces. The proof that copies of these prototiles can be put together to form a tiling of the plane, but cannot do so periodically, uses a construction that can be cast as a substitution tiling of the prototiles. In 1977 Robert Ammann discovered a number of sets of aperiodic prototiles, i.e.
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Schützenberger, Marcel-Paul, "On finite monoids having only trivial subgroups," Information and Control, Vol 8 No. 2, pp. 190–194, 1965. A consequence of the Krohn–Rhodes theorem is that every finite aperiodic monoid divides a wreath product of copies of the three- element flip-flop monoid, consisting of an identity element and two right zeros. The two-sided Krohn–Rhodes theorem alternatively characterizes finite aperiodic monoids as divisors of iterated block products of copies of the two- element semilattice.
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Substitution tiling systems provide a rich source of aperiodic tilings. A set of tiles that forces a substitution structure to emerge is said to enforce the substitution structure. For example, the chair tiles shown below admit a substitution, and a portion of a substitution tiling is shown at right below. These substitution tilings are necessarily non- periodic, in precisely the same manner as described above, but the chair tile itself is not aperiodic—it is easy to find periodic tilings by unmarked chair tiles.
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The Krohn–Rhodes complexity (also called group complexity or just complexity) of a finite semigroup S is the least number of groups in a wreath product of finite groups and finite aperiodic semigroups of which S is a divisor. All finite aperiodic semigroups have complexity 0, while non-trivial finite groups have complexity 1. In fact, there are semigroups of every non-negative integer complexity. For example, for any n greater than 1, the multiplicative semigroup of all (n+1)×(n+1) upper-triangular matrices over any fixed finite field has complexity n (Kambites, 2007).
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He calls this an aperiodic crystal. Its aperiodic nature allows it to encode an almost infinite number of possibilities with a small number of atoms. He finally compares this picture with the known facts and finds it in accordance with them. In chapter VI Schrödinger states: > ...living matter, while not eluding the "laws of physics" as established up > to date, is likely to involve "other laws of physics" hitherto unknown, > which however, once they have been revealed, will form just as integral a > part of science as the former.
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Frequently the term aperiodic was just used vaguely to describe the structures under consideration, referring to physical aperiodic solids, namely quasicrystals, or to something non-periodic with some kind of global order. The use of the word "tiling" is problematic as well, despite its straightforward definition. There is no single Penrose tiling, for example: the Penrose rhombs admit infinitely many tilings (which cannot be distinguished locally). A common solution is to try to use the terms carefully in technical writing, but recognize the widespread use of the informal terms.
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A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two-, three-, four-, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders—for instance, five- fold. Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of natural quasicrystals. The discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography.
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Penrose had discovered two simple sets of aperiodic tiles, each consisting of just two quadrilaterals. Since Penrose was taking out a patent, he wasn't ready to publish them, and Gardner's description was rather vague. Ammann wrote a letter to Gardner, describing his own work, which duplicated one of Penrose's sets, plus a foursome of "golden rhombohedra" that formed aperiodic tilings in space."The Mysterious Mr. Ammann" The Mathematical Intelligencer, September 2004, Volume 26, Issue 4, pp 10–21 More letters followed, and Ammann became a correspondent with many of the professional researchers.
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The first specific occurrence of aperiodic tilings arose in 1961, when logician Hao Wang tried to determine whether the Domino Problem is decidable — that is, whether there exists an algorithm for deciding if a given finite set of prototiles admits a tiling of the plane. Wang found algorithms to enumerate the tilesets that cannot tile the plane, and the tilesets that tile it periodically; by this he showed that such a decision algorithm exists if every finite set of prototiles that admits a tiling of the plane also admits a periodic tiling. In 1964 Robert Berger found an aperiodic set of prototiles from which he demonstrated that the tiling problem is in fact not decidable.. This first such set, used by Berger in his proof of undecidability, required 20,426 Wang tiles. Berger later reduced his set to 104, and Hans Läuchli subsequently found an aperiodic set requiring only 40 Wang tiles.
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Aperiodic multi-layers (chirped mirrors) emerging from a collaboration of Ferenc Krausz and Robert SzipöcsR. Szipöcs, K. Ferencz, Ch. Spielmann & F. Krausz, Opt. Lett. 19, 201 (1994). made such a control possible and are indispensable in today's femtosecond laser systems.
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An alt= The subject of aperiodic tilings received new interest in the 1960s when logician Hao Wang noted connections between decision problems and tilings. In particular, he introduced tilings by square plates with colored edges, now known as Wang dominoes or tiles, and posed the "Domino Problem": to determine whether a given set of Wang dominoes could tile the plane with matching colors on adjacent domino edges. He observed that if this problem were undecidable, then there would have to exist an aperiodic set of Wang dominoes. At the time, this seemed implausible, so Wang conjectured no such set could exist.
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Aperiodic tilings were considered as mathematical artefacts until 1984, when physicist Dan Shechtman announced the discovery of a phase of an aluminium- manganese alloy which produced a sharp diffractogram with an unambiguous fivefold symmetry – so it had to be a crystalline substance with icosahedral symmetry. In 1975 Robert Ammann had already extended the Penrose construction to a three-dimensional icosahedral equivalent. In such cases the term 'tiling' is taken to mean 'filling the space'. Photonic devices are currently built as aperiodical sequences of different layers, being thus aperiodic in one direction and periodic in the other two.
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In order that the quasicrystal itself be aperiodic, this slice must avoid any lattice plane of the higher-dimensional lattice. De Bruijn showed that Penrose tilings can be viewed as two-dimensional slices of five-dimensional hypercubic structures. Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors (the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice). The intuitive considerations obtained from simple model aperiodic tilings are formally expressed in the concepts of Meyer and Delone sets.
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Since the center-tap causes the signal from both vertical sections to be balanced, it creates a signal similar to a single vertical mast. When used with aperiodic windings, the sense circuit has to be wired into the receiver side, along with the tuning capacitor.
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Block and Weinberger used homological methods to construct aperiodic sets of tiles for all non-amenable manifolds. Joshua Socolar also gave another way to enforce aperiodicity, in terms of alternating condition. This generally leads to much smaller tile sets than the one derived from substitutions.
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To date, there is not a formal definition describing when a tiling has a hierarchical structure; nonetheless, it is clear that substitution tilings have them, as do the tilings of Berger, Knuth, Läuchli and Robinson. As with the term "aperiodic tiling" itself, the term "aperiodic hierarchical tiling" is a convenient shorthand, meaning something along the lines of "a set of tiles admitting only non-periodic tilings with a hierarchical structure". Each of these sets of tiles, in any tiling they admit, forces a particular hierarchical structure. (In many later examples, this structure can be described as a substitution tiling system; this is described below).
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Periodic structures lead to non-uniform current densities that lower efficiency and decrease stability. The aperiodic structure is typically made of either aerogels or somewhat more dense ambigels that forms a porous aperiodic sponge. Aerogels and ambigels are formed from wet gels; aerogels are formed when wet gels are dried such that no capillary forces are established, while ambigels are wet gels dried under conditions that minimize capillary forces. Aerogels and ambigels are unique in that 75-99% of the material is ‘open’ but interpenetrated by a solid that is on the order of 10 nm, resulting in pores on the order of 10 to 100 nm.
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In 1998, Madden and Aimee Johnson won the George Pólya Award for their joint paper on aperiodic tiling, "Putting the Pieces Together: Understanding Robinson's Nonperiodic Tilings". In 2017, Madden, Johnson, and their co-author Ayşe Şahin published the textbook Discovering Discrete Dynamical Systems through the Mathematical Association of America.
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Sakarovitch (2009) p.171 Regular languages of star-height 0 are also known as star-free languages. The theorem of Schützenberger provides an algebraic characterization of star-free languages by means of aperiodic syntactic monoids. In particular star-free languages are a proper decidable subclass of regular languages.
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Defining an aperiodic tiling (the pinwheel tiling) by repeatedly dissecting and inflating a rep-tile. Every square, rectangle, parallelogram, rhombus, or triangle is rep-4. The sphinx hexiamond (illustrated above) is rep-4 and rep-9, and is one of few known self-replicating pentagons. The Gosper island is rep-7.
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Unlike the kernels in traditional operating systems, S.Ha.R.K. is fully modular in terms of scheduling policies, aperiodic servers, and concurrency control protocols. Modularity is achieved by partitioning system activities between a generic kernel and a set of modules, which can be registered at initialization to configure the kernel according to specific application requirements.
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Amman's A and B pair of A5 tiles, decorated with matching rules; any tiling by these tilings is necessarily non- periodic, and the tiles are therefore aperiodic. Ammann A5 substitution rules, used to prove that the A5 tiles can only form non-periodic hierarchical tilings and thus are aperiodic tiles. This tiling exists in a 2D orthogonal projection of a 4D 8-8 duoprism constructed from 16 octahedral prisms. Amman's A and B tiles in his pair A5 a 45-135-degree rhombus and a 45-45-90 degree triangle, decorated with matching rules that allowed only certain arrangements in each region, forcing the non-periodic, hierarchical, and quasiperiodic structures of each of the infinite number of individual Ammann–Beenker tilings.
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A regular language is star-free if and only if it is accepted by an automaton with a finite and aperiodic transition monoid. This result of algebraic automata theory is due to Marcel-Paul Schützenberger. In particular, the minimum automaton of a star-free language is always counter-free (however, a star-free language may also be recognized by other automata which are not aperiodic). A counter-free language is a regular language for which there is an integer n such that for all words x, y, z and integers m ≥ n we have xymz in L if and only if xynz in L. Another way to state Schützenberger's theorem is that star-free languages and counter-free languages are the same thing.
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These tilings displayed instances of fivefold symmetry. One year later Alan Mackay showed experimentally that the diffraction pattern from the Penrose tiling had a two-dimensional Fourier transform consisting of sharp 'delta' peaks arranged in a fivefold symmetric pattern. Around the same time, Robert Ammann created a set of aperiodic tiles that produced eightfold symmetry.
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A Conus textile shell similar in appearance to Rule 30. Rule 30 is an elementary cellular automaton introduced by Stephen Wolfram in 1983. Using Wolfram's classification scheme, Rule 30 is a Class III rule, displaying aperiodic, chaotic behaviour. This rule is of particular interest because it produces complex, seemingly random patterns from simple, well-defined rules.
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In this way, setisets can produce non-periodic tilings. However, none of the non-periodic tilings thus far discovered qualify as aperiodic, because the prototiles can always be rearranged so as to yield a periodic tiling. Figure 5 shows the first two stages of inflation of an order 4 set leading to a non-periodic tiling.
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The smallest set of aperiodic tiles was discovered by Emmanuel Jeandel and Michael Rao in 2015, with 11 tiles and 4 colors. They used an exhaustive computer search to prove that 10 tiles or 3 colors are insufficient to force aperiodicity. This set, shown above in the title image, can be examined more closely at :File:Wang 11 tiles.svg.
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A periodic function is a function that repeats its values at regular intervals, for example, the trigonometric functions, which repeat at intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic. An illustration of a periodic function with period P.
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In quantum many-body systems, ground states of gapped Hamiltonians have exponential decay of correlations. In 2015 it was shown that the problem of determining the existence of a spectral gap is undecidable. The authors used an aperiodic tiling of quantum Turing machines and showed that this hypothetical material becomes gapped if and only if it halts.
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Conversely, planes that are not lattice planes have aperiodic intersections with the lattice called quasicrystals; this is known as a "cut- and-project" construction of a quasicrystal (and is typically also generalized to higher dimensions).J. B. Suck, M. Schreiber, and P. Häussler, eds., Quasicrystals: An Introduction to Structure, Physical Properties, and Applications (Springer: Berlin, 2004).
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What Is Life?, which conjectured genes were an "aperiodic crystal" storing codescript and influenced Francis Crick and James D. Watson in their 1953 identification of cellular DNA's molecular structure as a double helix.M. P. Murphy and L. A. J. O'Neill (1997). What Is Life? the Next Fifty Years: Speculations on the Future of Biology. Cambridge University Press. p 2.
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It is worth noting that there can be no aperiodic set of tiles in one dimension: it is a simple exercise to show that any set of tiles in the line either cannot be used to form a complete tiling, or can be used to form a periodic tiling. Aperiodicity of prototiles requires two or more dimensions.
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The human ear can recognize single sine waves as sounding clear because sine waves are representations of a single frequency with no harmonics. To the human ear, a sound that is made of more than one sine wave will have perceptible harmonics; addition of different sine waves results in a different waveform and thus changes the timbre of the sound. Presence of higher harmonics in addition to the fundamental causes variation in the timbre, which is the reason why the same musical note (the same frequency) played on different instruments sounds different. On the other hand, if the sound contains aperiodic waves along with sine waves (which are periodic), then the sound will be perceived to be noisy, as noise is characterized as being aperiodic or having a non-repetitive pattern.
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In 1963, Per-Olov Löwdin published proton tunneling as another mechanism for DNA mutation. In his paper, he stated that there is a new field of study called "quantum biology".Lowdin, P.O. (1965) Quantum genetics and the aperiodic solid. Some aspects on the Biological problems of heredity, mutations, aging and tumours in view of the quantum theory of the DNA molecule.
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Taylor and Socolar remark that the 3D monotile aperiodically tiles three-dimensional space. However the tile does allow tilings with a period, shifting one (non-periodic) two dimensional layer to the next, and so the tile is only "weakly aperiodic". Physical copies of the three-dimensional tile could not be fitted together without allowing reflections, which would require access to four-dimensional space.
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In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.
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The unexpected existence of aperiodic tilings, although not Berger's explicit construction of them, follows from another result proved by Berger: that the so-called domino problem is undecidable, disproving a conjecture of Hao Wang, Berger's advisor. The result is analogous to a 1962 construction used by Kahr, Moore, and Wang, to show that a more constrained version of the domino problem was undecidable.
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A patch of 25 monotiles, showing the triangular hierarchical structure The Socolar–Taylor tile is a single non-connected tile which is aperiodic on the Euclidean plane, meaning that it admits only non-periodic tilings of the plane (due to the Sierpinski's triangle-like tiling that occurs), with rotations and reflections of the tile allowed.. It is the first known example of a single aperiodic tile, or "einstein". The basic version of the tile is a simple hexagon, with printed designs to enforce a local matching rule, regarding how the tiles may be placed. It is currently unknown whether this rule may be geometrically implemented in two dimensions while keeping the tile a connected set. This is, however, confirmed to be possible in three dimensions, and, in their original paper, Socolar and Taylor suggest a three-dimensional analogue to the monotile.
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A rep-tile is labelled rep-n if the dissection uses n copies. Such a shape necessarily forms the prototile for a tiling of the plane, in many cases an aperiodic tiling. A rep- tile dissection using different sizes of the original shape is called an irregular rep-tile or irreptile. If the dissection uses n copies, the shape is said to be irrep-n.
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Aperiodic fibers are a subclass of Fresnel fibers which describe optical propagation in analogous terms to diffraction free beams. These too can be made by using air channels appropriately positioned on the virtual zones of the optical fiber. Photonic crystal fibers are a variant of the microstructured fibers reported by Kaiser et al. They are an attempt to incorporate the bandgap ideas of Yeh et al.
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Sometimes completing a transaction after its deadline has expired may be harmful or not helpful, and both the firm and hard deadlines consider this. An example of a firm deadline is an autopilot system. ;Soft deadline: If meeting time constrains is desirable but missing deadlines do not cause serious damage, a soft deadline may be best. It operates on an aperiodic or irregular schedule.
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The amount of confinement is currently unknown as only some models and experimental data exist. As with a previous method, the effects on the electrical conductivity have to be considered. Attempts to localize long- wavelength phonons by aperiodic superlattices or composite superlattices with different periodicities have been made. In addition, defects, especially dislocations, can be used to reduce thermal conductivity in low dimensional systems.
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This is reinforced by the purveyors of AP membranes; they are often sold with an electronic processor which, via equalization, restores the bass output lost through the mechanical damping. The effect of the equalization is opposite to that of the AP membrane, resulting in a loss of damping and an effective response similar to that of the loudspeaker without the aperiodic membrane and electronic processor.
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Mixed clays that have unequal components with random stacking produce aperiodic 00l diffraction patterns known as irrational patterns. The coefficient of variation (CV) is the percent standard deviation of the average of d(001) calculated from various reflections. If CV is less than 0.75% then the mineral is given a unique name. If CV is greater than 0.75% then mixed-layered nomenclature is used.
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The Wildmoossee The Wildmoos area is a large aquatic ecotope of about Biotopinventar Tirol: FSON Wildmoos, Lottenseegeb. Wildmoosgeb., 538,579.54 m² with two rarely appearing aperiodic lakes, the Lottensee and the Wildmoossee,Naturphänomen Wildmoossee und Lottensee - Tirol, tirol.at with a further bog and wetland area on the Wildalm.FMOOR Wildmoosalm 70.721,04 m², FSON Gebiet um Wildmoosalm - Koellental 63.321,02 m² The two areas are separated by forest.
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In the study of aperiodic flows, sampling at nearly periodic times, as for example, t_n= n+\varepsilon (n), where \varepsilon is positive and tends to zero, does not lead to a.e. convergence; in fact strong sweeping out occurs. This shows the possibility of serious errors when using the ergodic theorem for the study of physical systems. Such results can be of practical value for statisticians and other scientists.
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High speed digital imaging of the vocal folds (videokymography), another imaging technique, is not subject to the same limitations as laryngeal stroboscopy. A rigid endoscope is used to take images at a rate of 8000 frames per second, and the image is displayed in real time. As well, this technique allows imaging of aperiodic vibrations and can thus be used with patients presenting with all severities of dysphonia.
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They are remarkable by containing micrometre-sized grains of icosahedrite, a naturally occurring quasicrystal – aperiodic, yet ordered structure. The quasicrystal has a composition of Al63Cu24Fe13 and icosahedral symmetry. The presence of unoxidized aluminium in cupalite and association with the stishovite – a form of quartz which forms exclusively at high pressures of several tens GPa – suggest that cupalite is formed either upon meteoritic impact or in the deep earth mantle.
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Sound sources refer to the conversion of aerodynamic energy into acoustic energy. There are two main types of sound sources in the articulatory system: periodic (or more precisely semi-periodic) and aperiodic. A periodic sound source is vocal fold vibration produced at the glottis found in vowels and voiced consonants. A less common periodic sound source is the vibration of an oral articulator like the tongue found in alveolar trills.
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This notation indicates differing pitch, dynamics, articulation, instrumentation, timbre, and rhythm (duration and onset/order). Traditionally in Western music, a musical tone is a steady periodic sound. A musical tone is characterized by its duration, pitch, intensity (or loudness), and timbre (or quality). The notes used in music can be more complex than musical tones, as they may include aperiodic aspects, such as attack transients, vibrato, and envelope modulation.
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Such patterned surfaces can be used for different applications including tribology (wear and friction reduction), photovoltaics (increased photocurrent), or biotechnology. Electron interference lithography may be used for patterns which normally take too long for conventional electron beam lithography to generate. The drawback of interference lithography is that it is limited to patterning arrayed features or uniformly distributed aperiodic patterns only. Hence, for drawing arbitrarily shaped patterns, other photolithography techniques are required.
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The collected series of images is then reconstructed to a 3D volume by registering the image stack and removing artifacts. The predominant artifact that degrades FIB tomography is ion mill curtaining, where mill patterns form large aperiodic stripes in each image. The ion mill curtaining can be removed using destriping algorithms. FIB tomography can be done at both room and cryo temperatures as well as on both materials and biological samples.
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The Wildmoossee is an aperiodic mountain lake, 3 kilometres west of Seefeld in Tirol near the village of Wildmoos in the market borough of Telfs. The lake lies in the area of the water-soluble main dolomite of the Seefeld Plateau at a height of 1,316 metres.Naturphänomen Wildmoossee und Lottensee - Tirol, tirol.at As a result the ground underneath contains chasms that reach up to the bottom of the lake.
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The common justification for this argument, for example, according to renowned chemical engineer Kenneth Denbigh in his 1955 book The Principles of Chemical Equilibrium, is that "living organisms are open to their environment and can build up at the expense of foodstuffs which they take in and degrade." Schrödinger asked the question: "How does the living organism avoid decay?" The obvious answer is: "By eating, drinking, breathing and (in the case of plants) assimilating." While energy from nutrients is necessary to sustain an organism's order, Schrödinger also presciently postulated the existence of other molecules equally necessary for creating the order observed in living organisms: "An organism's astonishing gift of concentrating a stream of order on itself and thus escaping the decay into atomic chaos – of drinking orderliness from a suitable environment – seems to be connected with the presence of the aperiodic solids..." We now know that this "aperiodic" crystal is DNA, and that its irregular arrangement is a form of information.
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Light curves can be periodic, as in the case of eclipsing binaries, Cepheid variables, other periodic variables, and transiting extrasolar planets, or aperiodic, like the light curve of a nova, a cataclysmic variable star, a supernova or a microlensing event or binary as observed during occultation events. The study of the light curve, together with other observations, can yield considerable information about the physical process that produces it or constrain the physical theories about it.
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Wang tiles can be generalized in various ways, all of which are also undecidable in the above sense. For example, Wang cubes are equal-sized cubes with colored faces and side colors can be matched on any polygonal tessellation. Culik and Kari have demonstrated aperiodic sets of Wang cubes.. Winfree et al. have demonstrated the feasibility of creating molecular "tiles" made from DNA (deoxyribonucleic acid) that can act as Wang tiles.. Mittal et al.
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Aperiodic frequency is the rate of incidence or occurrence of non-cyclic phenomena, including random processes such as radioactive decay. It is expressed in units of measurement of reciprocal seconds (s−1) or, in the case of radioactivity, becquerels. It is defined as a ratio, f = N/T, involving the number of times an event happened (N) during a given time duration (T); it is a physical quantity of type temporal rate.
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Non-periodic tilings can also be obtained by projection of higher-dimensional structures into spaces with lower dimensionality and under some circumstances there can be tiles that enforce this non-periodic structure and so are aperiodic. The Penrose tiles are the first and most famous example of this, as first noted in the pioneering work of de Bruijn.N. G. de Bruijn, Nederl. Akad. Wetensch. Indag. Math. 43, 39–52, 53–66 (1981).
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In mathematics, the Rokhlin lemma, or Kakutani–Rokhlin lemma is an important result in ergodic theory. It states that an aperiodic measure preserving dynamical system can be decomposed to an arbitrary high tower of measurable sets and a remainder of arbitrarily small measure. It was proven by Vladimir Abramovich Rokhlin and independently by Shizuo Kakutani. The lemma is used extensively in ergodic theory, for example in Ornstein theory and has many generalizations.
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The road coloring problem is the problem of labeling the edges of a regular directed graph with the symbols of a k-letter input alphabet (where k is the outdegree of each vertex) in order to form a synchronizable DFA. It was conjectured in 1970 by Benjamin Weiss and Roy Adler that any strongly connected and aperiodic regular digraph can be labeled in this way; their conjecture was proven in 2007 by Avraham Trahtman..
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The mathematical counterpart of physical diffraction is the Fourier transform and the qualitative description of a diffraction picture as 'clear cut' or 'sharp' means that singularities are present in the Fourier spectrum. There are different methods to construct model quasicrystals. These are the same methods that produce aperiodic tilings with the additional constraint for the diffractive property. Thus, for a substitution tiling the eigenvalues of the substitution matrix should be Pisot numbers.
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One possible combination of source and filter in the human vocal tract. In human speech production, the sound source is the vocal folds, which can produce a periodic sound when constricted or an aperiodic (white noise) sound when relaxed. The filter is the rest of the vocal tract, which can change shape through manipulation of the pharynx, mouth, and nasal cavity. Fant roughly compares the source and filter to phonation and articulation, respectively.
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The missing CO detection is explained with two possible scenarios: Either dust grains are released in a collisional cascade induced by the collisions of km-sized planetesimals or a recent collision of planetary bodies generated a large amount of small dust grains. A light curve from CTIO shows variations, which could be disk material blocking light from the star. The TESS light curve shows aperiodic dipping on timescales of 0.5–2 days.
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Take a strongly connected and aperiodic social network. In this case, the common limiting belief is determined by the initial beliefs through : p(\infty) = s \cdot p(0) where s is the unique unit length left eigenvector of T corresponding to the eigenvalue 1. The vector s shows the weights that agents put on each other's initial beliefs in the consensus limit. Thus, the higher is s_i , the more influence individual i has on the consensus belief.
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In combinatorial mathematics, the necklace polynomial, or Moreau's necklace- counting function, introduced by , counts the number of distinct necklaces of n colored beads chosen out of α available colors. The necklaces are assumed to be aperiodic (not consisting of repeated subsequences), and the counting is done "without flipping over" (without reversing the order of the beads). This counting function describes, among other things, the number of free Lie algebras and the number of irreducible polynomials over a finite field.
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Aperiodic sound sources are the turbulent noise of fricative consonants and the short-noise burst of plosive releases produced in the oral cavity. Voicing is a common period sound source in spoken language and is related to how closely the vocal cords are placed together. In English there are only two possibilities, voiced and unvoiced. Voicing is caused by the vocal cords held close by each other, so that air passing through them makes them vibrate.
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György Buzsáki and his collaborators studied and characterized SWRs in detail and described their physiological functions and role in different states of the animal. These patterns are large amplitude, aperiodic recurrent oscillations occurring in the apical dendritic layer of the CA1 regions of the hippocampus. Sharp waves are followed by synchronous fast field oscillations (140–200 Hz frequency), named ripples. Features of these oscillations provided evidences for their role in inducing synaptic plasticity and memory consolidation.
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To ensure predictability, all aperiodic messages must be included in the bandwidth management calculations. Time Triggered Bus Scheduling ensures adequate flexibility for increasing network traffic during the lifetime of the system if growth potential is planned. As an example, system design will allow nodes to be integrated into the network without affecting the existing nodes. Furthermore, the predictable behavior enforced by Time Triggered Bus Scheduling allows systems with different criticality levels to coexist on the same network.
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Aperiodic multilayers have been proposed to reduce the sensitivity at the cost of lower reflectivity but are too sensitive to random fluctuations of layer thicknesses, such as from thickness control imprecision or interdiffusion. In particular, defocused dense lines at pitches up to twice the minimum resolvable pitch suffer wavelength-dependent edge shifts.Chromatic Blur in EUV Lithography A narrower bandwidth would increase sensitivity to mask absorber and buffer thickness on the 1 nm scale.M. Sugawara et al.
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Shahar Mozes (שחר מוזס) is an Israeli mathematician. Mozes received in 1991 his doctorate from the Hebrew University of Jerusalem with thesis Actions of Cartan subgroups under the supervision of Hillel Fürstenberg. (doctoral dissertation) At the Hebrew University of Jerusalem, Mozes became in 1993 a senior lecturer, in 1996 associate professor, and in 2002 a full professor. Moses does research on Lie groups and discrete subgroups of Lie groups, geometric group theory, ergodic theory, and aperiodic tilings.
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These coverings have been considered as a realistic model for the growth of quasicrystals: the overlapping decagons are 'quasi-unit cells' analogous to the unit cells from which crystals are constructed, and the matching rules maximize the density of certain atomic clusters.; see also The aperiodic nature of the coverings can make theoretical studies of physical properties, such as electronic structure, difficult due to the absence of Bloch's theorem. However, spectra of quasicrystals can still be computed with error control.
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A set S of finite binary words is balanced if for each n the subset Sn of words of length n has the property that the Hamming weight of the words in Sn takes at most two distinct values. A balanced sequence is one for which the set of factors is balanced. A balanced sequence has at most n+1 distinct factors of length n. An aperiodic sequence is one which does not consist of a finite sequence followed by a finite cycle.
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A Fibonacci crystal or quasicrystal is a model used to study systems with aperiodic structure. Both names are acceptable as a 'Fibonacci crystal' denotes a quasicrystal and a 'Fibonacci' quasicrystal is a specific type of quasicrystal. Fibonacci 'chains' or 'lattices' are closely related terms, depending on the dimension of the model.Searching database of physical papers reveals that 'Fibonacci chain' is the most frequently used; for instance more than 150 such items are found in Arxiv and just a few dozen other appelations.
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The forced neon bulb oscillator was the first system in which chaotic behavior was observed. Van der Pol and van der Mark wrote, concerning their experiments with demultiplication, that > Often an irregular noise is heard in the telephone receivers before the > frequency jumps to the next lower value. However this is a subsidiary > phenomenon, the main effect being the regular frequency demultiplication. Any periodic oscillation would have produced a musical tone; only aperiodic, chaotic oscillations would produce an "irregular noise".
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The Fourier–Bessel series expansion employs aperiodic and decaying Bessel functions as the basis. The Fourier–Bessel series expansion has been successfully applied in diversified areas such as Gear fault diagnosis, discrimination of odorants in a turbulent ambient, postural stability analysis, detection of voice onset time, glottal closure instants (epoch) detection, separation of speech formants, EEG signal segmentation, speech enhancement, and speaker identification. The Fourier–Bessel series expansion has also been used to reduce cross terms in the Wigner–Ville distribution.
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A single 30-tetrahedron ring Boerdijk–Coxeter helix within the 600-cell, seen in stereographic projection Regular tetrahedra can be stacked face-to-face in a chiral aperiodic chain called the Boerdijk–Coxeter helix. In four dimensions, all the convex regular 4-polytopes with tetrahedral cells (the 5-cell, 16-cell and 600-cell) can be constructed as tilings of the 3-sphere by these chains, which become periodic in the three-dimensional space of the 4-polytope's boundary surface.
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A 5-cube as an orthographic projection into 2D using Petrie polygon basis vectors overlaid on the diffractogram from an icosahedral Ho-Mg-Zn quasicrystal A 6-cube projected into the rhombic triacontahedron using the golden ratio in the basis vectors. This is used to understand the aperiodic icosahedral structure of quasicrystals. There are several ways to mathematically define quasicrystalline patterns. One definition, the "cut and project" construction, is based on the work of Harald Bohr (mathematician brother of Niels Bohr).
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The usual way to solve such a system is to first diagonalize the W matrix. Its diagonal entries will be eigenvalues corresponding to certain linear combinations of certain subsets of sequences which will be eigenvectors of the W matrix. These subsets of sequences are the quasispecies. Assuming that the matrix W is a primitive matrix (irreducible and aperiodic), then after very many generations only the eigenvector with the largest eigenvalue will prevail, and it is this quasispecies that will eventually dominate.
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The above variations can be thought of as a two- dimensional square, because the world is two-dimensional and laid out in a square grid. One-dimensional square variations, known as elementary cellular automata, and three-dimensional square variations have been developed, as have two-dimensional hexagonal and triangular variations. A variant using aperiodic tiling grids has also been made. Conway's rules may also be generalized such that instead of two states, live and dead, there are three or more.
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More generally, a Markov chain is ergodic if there is a number N such that any state can be reached from any other state in any number of steps less or equal to a number N. In case of a fully connected transition matrix, where all transitions have a non-zero probability, this condition is fulfilled with N = 1\. A Markov chain with more than one state and just one out-going transition per state is either not irreducible or not aperiodic, hence cannot be ergodic.
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J. Canning, Fresnel Optics Inside Optical Fibres, in Photonics Research Developments, Chapter 5, Nova Science Publishers, United States, (2008) and refs therein These Fresnel fibers use well known Fresnel optics which has long been applied to lens design, including more advanced forms used in aperiodic, fractal, and irregular adaptive optics, or Fresnel/fractal zones. Many other practical design benefits include broader photonic bandgaps in diffraction based propagating waveguides and reduced bend losses, important for achieving structured optical fibers with propagation losses below that of step-index fibers.
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While it is used mostly as a theoretical construct, physical models are implemented in order to verify the concept empirically. Most of its applications pertain to various areas of solid state physics. Two-dimensional aperiodic tiling based on the Fibonacci word The mathematical properties of the Fibonacci word and related topics are well researched and readily applied in such studies. The elements of a Fibonacci crystal structure are arranged in one or more spatial dimensions according to the sequence given by the Fibonacci word.
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The aperiodic nature of quasicrystals can also make theoretical studies of physical properties, such as electronic structure, difficult due to the inapplicability of Bloch's theorem. However, spectra of quasicrystals can still be computed with error control. Study of quasicrystals may shed light on the most basic notions related to the quantum critical point observed in heavy fermion metals. Experimental measurements on the gold- aluminium-ytterbium quasicrystal have revealed a quantum critical point defining the divergence of the magnetic susceptibility as temperature tends to zero.
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Irreducible polynomials over finite fields are also useful for Pseudorandom number generators using feedback shift registers and discrete logarithm over F2n. The number of irreducible monic polynomials of degree n over Fq is the number of aperiodic necklaces, given by Moreau's necklace- counting function Mq(n). The closely related necklace function Nq(n) counts monic polynomials of degree n which are primary (a power of an irreducible); or alternatively irreducible polynomials of all degrees d which divide n.Christophe Reutenauer, Mots circulaires et polynomes irreductibles, Ann. Sci.
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Jean-Pierre Gazeau (born 10 October 1945) is a French physicist and mathematician who works in the field of symmetry in quantum physics.. His research has focused on coherent states; beta numeration for quasicrystals, and more generally for aperiodic order; and de Sitter space and anti-de Sitter space times.He is a professor emeritus at Paris Diderot University, Sorbonne Paris Cité University (group).Through a career spanning 50 years, he has held research and teaching positions on five continents, with a particular concentration on developing and emerging countries.
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Even minor changes in the weather, physical layout or even bumping the chassis containing the tunable capacitors can cause the tuning to vary. For this reason a variety of systems were used to decrease the sensitivity of the radiogoniometer to mis- tuning. Primary among these was the aperiodic aerial concept, which described the mechanical layout of the radiogoniometer's internal wiring. By winding the sense coil wiring around a vertical cylinder, and wiring the field coils in a similar arrangement as close as possible to the sense coil, the entire circuit became capacitively coupled.
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DX arrays: DX arrays: Two-dimensional arrays have been made from other motifs as well, including the Holliday junction rhombus lattice,Other arrays: and various DX-based arrays making use of a double- cohesion scheme.Other arrays: Other arrays: The top two images at right show examples of tile-based periodic lattices. Two-dimensional arrays can be made to exhibit aperiodic structures whose assembly implements a specific algorithm, exhibiting one form of DNA computing. The DX tiles can have their sticky end sequences chosen so that they act as Wang tiles, allowing them to perform computation.
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Box the speakers were packaged in; showing foam grille HD77 is a model name of Marantz 4-way high-fidelity loudspeakers which were produced during the mid-1970s. They were bass reflex speakers, but came with a cylindrical piece of foam which fit into the bass-reflex port of the enclosure if the listener preferred the more accurate bass response provided by airtight speaker boxes. However, they were not fully airtight this way, but rather what is known as aperiodic. They were designed by former JBL Engineer Edmond May.
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Patterns may be elaborated by the use of two levels of design, as at the 1453 Darb-e Imam shrine. Square repeating units of known patterns can be copied as templates, and historic pattern books may have been intended for use in this way. The 15th century Topkapı Scroll explicitly shows girih patterns together with the tilings used to create them. A set of tiles consisting of a dart and a kite shape can be used to create aperiodic Penrose tilings, though there is no evidence that such a set was used in medieval times.
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A lattice arrangement (commonly called a regular arrangement) is one in which the centers of the spheres form a very symmetric pattern which needs only n vectors to be uniquely defined (in n-dimensional Euclidean space). Lattice arrangements are periodic. Arrangements in which the spheres do not form a lattice (often referred to as irregular) can still be periodic, but also aperiodic (properly speaking non-periodic) or random. Lattice arrangements are easier to handle than irregular ones—their high degree of symmetry makes it easier to classify them and to measure their densities.
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It is also possible to restrict the classes of point sets that may be Danzer sets in other ways than by their densities. In particular, they cannot be the union of finitely many lattices, they cannot be generated by choosing a point in each tile of a substitution tiling (in the same position for each tile of the same type), and they cannot be generated by the cut-and-project method for constructing aperiodic tilings. Therefore, the vertices of the pinwheel tiling and Penrose tiling are not Danzer sets.
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The road coloring problem is the problem of edge-coloring a directed graph with uniform out- degrees, in such a way that the resulting automaton has a synchronizing word. solved the road coloring problem by proving that such a coloring can be found whenever the given graph is strongly connected and aperiodic. Ramsey's theorem concerns the problem of -coloring the edges of a large complete graph in order to avoid creating monochromatic complete subgraphs of some given size . According to the theorem, there exists a number such that, whenever , such a coloring is not possible.
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Dynaco was a primary producer of these enclosures for many years, using designs developed by a Scandinavian driver maker. The design remains uncommon among commercial designs currently available. A reason for this may be that adding damping material is a needlessly inefficient method of increasing damping; the same alignment can be achieved by simply choosing a loudspeaker driver with the appropriate parameters and precisely tuning the enclosure and port for the desired response. A similar technique has been used in aftermarket car audio; it is called "aperiodic membrane" (AP).
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The aperiodic structures obtained by the cut-and-project method are made diffractive by choosing a suitable orientation for the construction; this is a geometric approach that has also a great appeal for physicists. Classical theory of crystals reduces crystals to point lattices where each point is the center of mass of one of the identical units of the crystal. The structure of crystals can be analyzed by defining an associated group. Quasicrystals, on the other hand, are composed of more than one type of unit, so, instead of lattices, quasilattices must be used.
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It is commonly accepted that an implementation of a Fixed-priority pre-emptive scheduling (FPS) is simpler than a dynamic priority scheduler, like the EDF. However, when comparing the maximum usage of an optimal scheduling under fixed priority (with the priority of each thread given by the Rate-monotonic scheduling), the EDF can reach 100% while the theorical maximum value for Rate-monotonic scheduling is around 69%. Note that EDF does not make any specific assumption on the periodicity of the tasks; hence, it can be used for scheduling periodic as well as aperiodic tasks.
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In the case of rational polygons, all the orbits are periodic. In 1995, Tabachnikov showed that outer billiards for the regular pentagon has some aperiodic orbits, thus clarifying the distinction between the dynamics in the rational and regular cases. In 1996, Boyland showed that outer billiards relative to some shapes can have orbits which accumulate on the shape. In 2005, D. Genin showed that all orbits are bounded when the shape is a trapezoid, thus showing that quasirationality is not a necessary condition for the system to have all orbits bounded.
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Robert Ammann (October 1, 1946 – May, 1994) was an amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings. Ammann–Beenker tiling Ammann attended Brandeis University, but generally did not go to classes, and left after three years. He worked as a programmer for Honeywell. After ten years, his position was eliminated as part of a routine cutback, and Ammann ended up working as a mail sorter for a post office. In 1975, Ammann read an announcement by Martin Gardner of new work by Roger Penrose.
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Grünbaum and G.C. Shephard, Tilings and Patterns, Freemann, NY 1986 and later, in collaboration with the authors of the book, he published a paperR.Ammann, B. Grünbaum and G.C. Shephard, Aperiodic Tiles, Discrete Comput Geom 8 (1992),1–25 proving the aperiodicity for four of them. Ammann's discoveries came to notice only after Penrose had published his own discovery and gained priority. In 1981 de Bruijn exposed the cut and project method and in 1984 came the sensational news about Shechtman quasicrystals which promoted the Penrose tiling to fame.
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Example of Wang tessellation with 13 tiles. In 1961, Wang conjectured that if a finite set of Wang tiles can tile the plane, then there exists also a periodic tiling, i.e., a tiling that is invariant under translations by vectors in a 2-dimensional lattice, like a wallpaper pattern. He also observed that this conjecture would imply the existence of an algorithm to decide whether a given finite set of Wang tiles can tile the plane.. Wang proposes his tiles, and conjectures that there are no aperiodic sets.. Presents the domino problem for a popular audience.
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The result can be further improved by crystal tilt compensation and search for the most likely projected symmetry. In conclusion one can say that the exit-wave function reconstruction method has most advantages for determining the (aperiodic) atomic structure of defects and small clusters and CIP is the method of choice if the periodic structure is in focus of the investigation or when defocus series of HREM images cannot be obtained, e.g. due to beam damage of the sample. However, a recent study on the catalyst related material Cs0.5[Nb2.5W2.5O14] shows the advantages when both methods are linked in one study.
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In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components. These components correspond to finite aperiodic semigroups and finite simple groups that are combined together in a feedback-free manner (called a "wreath product" or "cascade"). Krohn and Rhodes found a general decomposition for finite automata. In doing their research, though, the authors discovered and proved an unexpected major result in finite semigroup theory, revealing a deep connection between finite automata and semigroups.
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Time-triggered systems can be viewed as a subset of a more general event-triggered (ET) system architecture (see event-driven programming). Implementation of an ET system will typically involve use of multiple interrupts, each associated with specific periodic events (such as timer overflows) or aperiodic events (such as the arrival of messages over a communication bus at random points in time). ET designs are traditionally associated with the use of what is known as a real-time operating system (or RTOS), though use of such a software platform is not a defining characteristic of an ET architecture.
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Schrödinger's lecture focused on one important question: "how can the events in space and time which take place within the spatial boundary of a living organism be accounted for by physics and chemistry?" In the book, Schrödinger introduced the idea of an "aperiodic crystal" that contained genetic information in its configuration of covalent chemical bonds. In the 1950s, this idea stimulated enthusiasm for discovering the genetic molecule. Although the existence of some form of hereditary information had been hypothesized since 1869, its role in reproduction and its helical shape were still unknown at the time of Schrödinger's lecture.
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In retrospect, Schrödinger's aperiodic crystal can be viewed as a well-reasoned theoretical prediction of what biologists should have been looking for during their search for genetic material. Both James D. Watson,. Page 28 details how Watson came to appreciate the significance of the gene. and Francis Crick, who jointly proposed the double helix structure of DNA based on, amongst other theoretical insights, X-ray diffraction experiments by Rosalind Franklin, credited Schrödinger's book with presenting an early theoretical description of how the storage of genetic information would work, and each independently acknowledged the book as a source of inspiration for their initial researches.
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In the limit the plane would be divided into girih tiles that repeat with frequencies that are aperiodic. The use of such a subdivision rule would serve as evidence that Islamic artisans of the 15th century were aware that girih tiles can produce complex patterns that never exactly repeat themselves. However, no known patterns made with girih tiles have more than a two-level design. There would have been no practical need for a girih pattern with more than two levels of design, as a third level would be either too large or too small to be perceived.
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Generally, such fibers are constructed by the same methods as other optical fibers: first, one constructs a "preform" on the scale of centimeters in size, and then heats the preform and draws it down to a much smaller diameter (often nearly as small as a human hair), shrinking the preform cross section but (usually) maintaining the same features. In this way, kilometers of fiber can be produced from a single preform. The most common method involves stacking, although drilling/milling was used to produce the first aperiodic designs. This formed the subsequent basis for producing the first soft glass and polymer structured fibers.
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Due to the secret passageway, the Markov chain is also aperiodic, because the monsters can move from any state to any state both in an even and in an uneven number of state transitions. Therefore, a unique stationary distribution exists and can be found by solving \pi Q=0, subject to the constraint that elements must sum to 1. The solution of this linear equation subject to the constraint is \pi=(7.7,15.4,7.7,11.5,15.4,11.5,7.7,15.4,7.7)\%. The central state and the border states 2 and 8 of the adjacent secret passageway are visited most and the corner states are visited least.
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In the superspace picture, aperiodic crystals are obtained from the section of a periodic crystal of higher dimension (up to 6D) cut at an irrational angle. While phonons change the position atoms relative to the crystal structure in space, phasons change the position of atoms relative to the quasi-crystal structure and the cut through superspace that defines it. Phonon modes are therefore excitations of the "in plane" real (also called parallel or external) space whereas phasons are excitations of the perpendicular (also called internal) space. The hydrodynamic theory of the quasicrystals predicts that the conventional (phonon) strain relaxes rapidly.
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This phenomenon is known as "submultiplication" or "demultiplication", and was first observed in 1927 by Balthasar van der Pol and his collaborator Jan van der Mark. In some cases the ratio of the external frequency to the frequency of the oscillation observed in the circuit may be a rational number, or even an irrational one (the latter case is known as the "quasiperiodic" regime). When the periodic and quasiperiodic regimes overlap, the behavior of the circuit may become aperiodic, meaning that the pattern of the oscillations never repeats. This aperiodicity correspond to the behavior of the circuit becoming chaotic (see chaos theory).
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The reason why planes where the components (in the reciprocal-lattice basis) have rational ratios are of special interest is that these are the lattice planes: they are the only planes whose intersections with the crystal are 2d-periodic. For a plane (abc) where a, b and c have irrational ratios, on the other hand, the intersection of the plane with the crystal is not periodic. It forms an aperiodic pattern known as a quasicrystal. This construction corresponds precisely to the standard "cut-and-project" method of defining a quasicrystal, using a plane with irrational-ratio Miller indices.
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This cell group normally consists of four clustered cells in either side of the brain, roughly halfway between the top and bottom edge, in the posterior area of the brain. Cells in this cluster are occasionally located abnormally near the top edge, rather than the middle, of the brain at a rate of about 17% of cells in wild-type D. melanogaster. The per01 mutation significantly increases the percentage of abnormally located cells to about 40%. In two aperiodic strains of D. pseudoobscura, the percentages of abnormally located cells are likewise significantly increased over those in the wild type.
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Mathematically, quasicrystals have been shown to be derivable from a general method that treats them as projections of a higher-dimensional lattice. Just as circles, ellipses, and hyperbolic curves in the plane can be obtained as sections from a three-dimensional double cone, so too various (aperiodic or periodic) arrangements in two and three dimensions can be obtained from postulated hyperlattices with four or more dimensions. Icosahedral quasicrystals in three dimensions were projected from a six- dimensional hypercubic lattice by Peter Kramer and Roberto Neri in 1984. The tiling is formed by two tiles with rhombohedral shape.
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Gathering for Gardner Celebration of Mind Presenter During 1995 he did research at The Geometry Center, a mathematics research and education center at the University of Minnesota, where he investigated aperiodic tilings of the plane.Chaim Goodman- Strauss: Activities at the Geometry Center University Of Minnesota Goodman- Strauss has been fascinated by patterns and mathematical paradoxes for as long as he can remember. He attended a lecture about the mathematician Georg Cantor when he was 17 and says, "I was already doomed to be a mathematician, but that lecture sealed my fate."The Shape of Everyday Things by Melissa Lutz Blouin.
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Dead individuals that washed up on the beaches were young, born in the previous autumn, and had full stomachs. Their appearance at such a high density was the result of an aperiodic outbreak in the Andes, and living specimens that were caught in traps showed no signs of being in breeding condition. It was not until the following year that numbers of individuals returned to their normal level for the area and breeding started taking place again. O. longicaudatus is the principal reservoir host of Andes virus (ANDV), which causes most cases of hantavirus cardiopulmonary syndrome in South America.
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This animation illustrates the process: An animated gif illustrating how later terms of the Kolakoski sequence are generated by earlier terms. These self- generating properties, which remain if the sequence is written without the initial 1, mean that the Kolakoski sequence can be described as a fractal, or mathematical object that encodes its own representation on other scales. Bertran Steinsky has created a recursive formula for the i-th term of the sequence but the sequence is believed to be aperiodic, that is, its terms do not have a general repeating pattern (cf. irrational numbers like π and ).
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In other words, the domino problem asks whether there is an effective procedure that correctly settles the problem for all given domino sets. In 1966, Berger solved the domino problem in the negative. He proved that no algorithm for the problem can exist, by showing how to translate any Turing machine into a set of Wang tiles that tiles the plane if and only if the Turing machine does not halt. The undecidability of the halting problem (the problem of testing whether a Turing machine eventually halts) then implies the undecidability of Wang's tiling problem.. Berger coins the term "Wang tiles", and demonstrates the first aperiodic set of them.
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The girih patterns on the Darb-e Imam shrine built in 1453 at Isfahan had a much more complex pattern than any previously seen. The details of the pattern indicate that girih tiles, rather than compass and straightedge, were used for decorating the shrine. The patterns appear aperiodic; within the area on the wall where they are displayed, they do not form a regularly repeating pattern; and they are drawn at two different scales. A large-scale pattern is discernible when the building is viewed from a distance, and a smaller-scale pattern forming part of the larger one can be seen from closer up.
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Unable to get her own faculty position at Arizona because of the anti-nepotism rules then in place, she and her husband visited Brazil, supported by a Fulbright Scholarship. They then moved to Massachusetts, where she took the faculty position at Smith that she would keep for the rest of her career. She eventually divorced Senechal, and married photographer Stan Sherer in 1989. She retired in 2007; a festival in 2006 honoring her impending retirement included the performance of a musical play that she wrote with The Talking Band member Ellen Maddow, loosely centered around the theme of aperiodic tilings and the life of amateur mathematician Robert Ammann...
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Further advances in acoustic phonetics were made possible by the development of the telephone industry. (Incidentally, Alexander Graham Bell's father, Alexander Melville Bell, was a phonetician.) During World War II, work at the Bell Telephone Laboratories (which invented the spectrograph) greatly facilitated the systematic study of the spectral properties of periodic and aperiodic speech sounds, vocal tract resonances and vowel formants, voice quality, prosody, etc. Integrated linear prediction residuals (ILPR) was an effective feature proposed by T V Ananthapadmanabha in 1995, which closely approximates the voice source signal.T. V. Ananthapadmanabha, "Acoustic factors determining perceived voice quality," in Vocal fold Physiology - Voice quality control, O.Fujimura and M. Hirano, Eds.
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Specific ideas from Penrose's work include: the idea that the human mind operates in certain fundamental ways as a quantum computer, espoused in Penrose's The Emperor's New Mind; Platonic realism as a philosophical basis for works of fiction, as in stories from Penrose's The Road to Reality; and the theory of aperiodic tilings, which appear in the Teglon puzzle in the novel. Stephenson also cites as an influence the works of Kurt Gödel and Edmund Husserl, both of whom the character Durand mentions by name in the novel. Much of the Geometers' technology seen in the novel reflects existing scientific concepts. The alien ship moves by means of nuclear pulse propulsion.
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Some movements are aperiodic, others regular, as the Earth tides caused by the lunar and solar gravitational field. The pendulums measure a distance of 95 m between the upper and lower mountings, which contributes to the fact that the instruments detect tectonic movements with high precision and are relatively immune to some of the noise which affects smaller instruments. The Earth's crust is the outer brittle layer of our planet, on average thick in continental areas. This moves up and down by ten centimeters during the time of 12 and 24 hours due to the attraction of Moon and Sun, and is accompanied by a local tilting of some parts in a billion of radians.
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These have modern equivalents such as kilohertz (kHz), megahertz (MHz), and gigahertz (GHz). Following the introduction of the SI standard, use of these terms began to fall off in favor of the new unit, with hertz becoming the dominant convention in both academic and colloquial speech by the 1970s. The rate at which aperiodic or stochastic events occur may be expressed in becquerels (as in the case of radioactive decay), not hertz, since although the two are mathematically similar, by convention hertz implies regularity where becquerels implies the requirement of a time averaging operation. Thus, one becquerel is one event per second on average, whereas one hertz is one event per second on a regular cycle.
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Wang tiles have recently become a popular tool for procedural synthesis of textures, heightfields, and other large and nonrepeating bidimensional data sets; a small set of precomputed or hand-made source tiles can be assembled very cheaply without too obvious repetitions and without periodicity. In this case, traditional aperiodic tilings would show their very regular structure; much less constrained sets that guarantee at least two tile choices for any two given side colors are common because tileability is easily ensured and each tile can be selected pseudorandomly.. Introduces the idea of using Wang tiles for texture variation, with a deterministic substitution system.. Introduces stochastic tiling.. . Applies Wang Tiles for real-time texturing on a GPU.. . Shows advanced applications.
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The Koch snowflake is irrep-7: six small snowflakes of the same size, together with another snowflake with three times the area of the smaller ones, can combine to form a single larger snowflake. A right triangle with side lengths in the ratio 1:2 is rep-5, and its rep-5 dissection forms the basis of the aperiodic pinwheel tiling. By Pythagoras' theorem, the hypotenuse, or sloping side of the rep-5 triangle, has a length of . The international standard ISO 216 defines sizes of paper sheets using the , in which the long side of a rectangular sheet of paper is the square root of two times the short side of the paper.
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Rustle noise is noise consisting of aperiodic pulses characterized by the average time between those pulses (such as the mean time interval between clicks of a Geiger counter), known as rustle time (Schouten ?). Rustle time is determined by the fineness of sand, seeds, or shot in rattles, contributes heavily to the sound of sizzle cymbals, drum snares, drum rolls, and string drums, and makes subtle differences in string instrument sounds. Rustle time in strings is affected by different weights and widths of bows and by types of hair and rosin in strings. The concept is also applicable to flutter-tonguing, brass and woodwind growls, resonated vocal fry in woodwinds, and eructation sounds in some woodwinds.
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A recent characterization of grid graphs having total perfect codes S (i.e. with just 1-cubes as induced components, also called 1-PDDS and DPL(2,4)), due to Klostermeyer and Goldwasser,Klostermeyer W. F.; Goldwasser J. L. "Total Perfect Codes in Grid Graphs", Bull. Inst. Comb. Appl., 46(2006) 61-68. allowed Dejter and Delgado to show that these sets S are restrictions of only one total perfect code S1 in the planar integer lattice graph, with the extra- bonus that the complement of S1 yields an aperiodic tiling, like the Penrose tiling. In contrast, the parallel, horizontal, total perfect codes in the planar integer lattice graph are in 1-1 correspondence with the doubly infinite {0,1}-sequences.
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In dynamical systems, Johnson is known for her work on a conjecture of Hillel Furstenberg on the classification of invariant measures for the action of two independent modular multiplication operations on an interval. In 1998, Johnson and Kathleen Madden won the George Pólya Award for their joint paper on aperiodic tiling, "Putting the Pieces Together: Understanding Robinson's Nonperiodic Tilings". In 2017, Madden, Johnson, and their co-author Ayşe Şahin published the textbook Discovering Discrete Dynamical Systems through the Mathematical Association of America. With Joseph Auslander and Cesar E. Silva she is also the co-editor of Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby (Contemporary Mathematics 678, American Mathematical Society, 2016).
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Another investigation of stochastic resonance in broadband (or, equivalently, aperiodic) signals was conducted by probing cutaneous mechanoreceptors in the rat. A patch of skin from the thigh and its corresponding section of the saphenous nerve were removed, mounted on a test stand immersed in interstitial fluid. Slowly adapting type 1 (SA1) mechanoreceptors output signals in response to mechanical vibrations below 500 Hz. The skin was mechanically stimulated with a broadband pressure signal with varying amounts of broadband noise using the up-and-down motion of a cylindrical probe. The intensity of the pressure signal was tested without noise and then set at a near sub-threshold intensity that would evoke 10 action potentials over a 60-second stimulation time.
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The IOD involves an aperiodic oscillation of sea-surface temperatures (SST), between "positive", "neutral" and "negative" phases. A positive phase sees greater- than-average sea-surface temperatures and greater precipitation in the western Indian Ocean region, with a corresponding cooling of waters in the eastern Indian Ocean—which tends to cause droughts in adjacent land areas of Indonesia and Australia. The negative phase of the IOD brings about the opposite conditions, with warmer water and greater precipitation in the eastern Indian Ocean, and cooler and drier conditions in the west. The IOD also affects the strength of monsoons over the Indian subcontinent. A significant positive IOD occurred in 1997–98, with another in 2006.
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A given set of tiles, in the Euclidean plane or some other geometric setting, admits a tiling if non- overlapping copies of the tiles in the set can be fitted together to cover the entire space. A given set of tiles might admit periodic tilings — that is, tilings which remain invariant after being shifted by a translation (for example, a lattice of square tiles is periodic). It is not difficult to design a set of tiles that admits non-periodic tilings as well as periodic tilings (for example, randomly arranged tilings using a 2×2 square and 2×1 rectangle will typically be non-periodic). However, an aperiodic set of tiles can only produce non-periodic tilings.
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Not all regions within a protein mutate at the same rate; functionally important areas mutate more slowly and amino acid substitutions involving similar amino acids occurs more often than dissimilar substitutions. Overall, the level of polymorphisms in proteins seems to be fairly constant. Several species (including humans, fruit flies, and mice) have similar levels of protein polymorphism. In his Dublin 1943 lectures, “What Is Life?”, Erwin Schrodinger proposed that we could progress in answering this question by using statistical mechanics and partition functions, but not quantum mechanics and his wave equation. He described an “aperiodic crystal” which could carry genetic information, a description credited by Francis Crick and James D. Watson with having inspired their discovery of the double helical structure of DNA .
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Periodic tiling by the sphinx With pentagons that are not required to be convex, additional types of tiling are possible. An example is the sphinx tiling, an aperiodic tiling formed by a pentagonal rep- tile. The sphinx may also tile the plane periodically, by fitting two sphinx tiles together to form a parallelogram and then tiling the plane by translates of this parallelogram, a pattern that can be extended to any non-convex pentagon that has two consecutive angles adding to 2, thus satisfying the condition(s) of convex Type 1 above. It is possible to divide an equilateral triangle into three congruent non-convex pentagons, meeting at the center of the triangle, and to tile the plane with the resulting three-pentagon unit.
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There are an infinite number of uniform tilings of the hyperbolic plane by kites, the simplest of which is the deltoidal triheptagonal tiling. Kites and darts in which the two isosceles triangles forming the kite have apex angles of 2π/5 and 4π/5 represent one of two sets of essential tiles in the Penrose tiling, an aperiodic tiling of the plane discovered by mathematical physicist Roger Penrose. Face-transitive self- tesselation of the sphere, Euclidean plane, and hyperbolic plane with kites occurs as uniform duals: for Coxeter group [p,q], with any set of p,q between 3 and infinity, as this table partially shows up to q=6. When p=q, the kites become rhombi; when p=q=4, they become squares.
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Tones were seen as analogous to vowels, and noises to consonants in human speech, and because traditional music had emphasised tones almost exclusively, composers of electronic music saw scope for exploration along the continuum stretching from single, pure (sine) tones to white noise (the densest superimposition of all audible frequencies)—that is, from entirely periodic to entirely aperiodic sound phenomena. In a process opposite to the building up of sine tones into complexes, white noise could be filtered to produce sounds with different bandwidths, called "coloured noises", such as the speech sounds represented in English by sh, f, s, or ch. An early example of an electronic composition composed entirely by filtering white noise in this way is Henri Pousseur's Scambi (Exchanges), realised at the Studio di Fonologia in Milan in 1957.
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The characteristics found in the light curve of VVV-WIT-07 (WIT refers to "What Is This?") are similar to those seen in J1407 (Mamajek's Object), a pre-MS K5 dwarf with a ring system that eclipses the star or, alternatively, to KIC 8462852 (or Tabby's star), a F3 IV/V star that shows irregular and aperiodic obscurations in its light curve. From 2010 to 2018, the star dimmed and brightened irregularly (v~14.35 – 16.164), and seemed similar to Tabby's star, except the light from VVV-WIT-07 dimmed by up to 80 percent, while Tabby’s star faded by only about 20 percent. Another star, J1407, however, has been found to have dimmed by up to 95%, which may be more similar to the light curve presented by VVV-WIT-07.
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X-ray diffraction pattern of the natural Al63Cu24Fe13 quasicrystal. Khatyrkite is remarkable in that it contains micrometre-sized grains of icosahedrite, the first known naturally occurring quasicrystal—aperiodic and yet ordered in structure. The quasicrystal has a composition of Al63Cu24Fe13 which is close to that of a well-characterized synthetic Al-Cu-Fe material. It is thought that the icosahedrite, like the khatyrkite, was formed in space in a collision involving the parent body of the meteorite. A second natural quasicrystal, called decagonite, Al71Ni24Fe5 with a decagonal structure has been identified by Luca Bindi in the samples and announced in 2015.Bindi L., and al, Natural quasicrystal with decagonal symmetry, Nature - Scientific Reports 5, Article number: 9111 doi:10.1038/srep09111Bindi, Luca, et al. "Decagonite, Al71Ni24Fe5, a quasicrystal with decagonal symmetry from the Khatyrka CV3 carbonaceous chondrite." American Mineralogist 100.10 (2015): 2340-2343.
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As such, photonic crystal fibers may be considered a subgroup of microstructured optical fibers. There are two main classes of MOF # Index guided fibers, where guiding is obtained through effect of total internal reflection # Photonic bandgap fibers, where guiding is obtained through constructive interference of scattered light (including photonic bandgap effect.) Structured optical fibers, those based on channels running along their entire length go back to Kaiser and Co in 1974. These include air-clad optical fibers, microstructured optical fibers sometimes called photonic crystal fiber when the arrays of holes are periodic and look like a crystal, and many other subclasses. Martelli and Canning realized that the crystal structures that have identical interstitial regions are actually not the most ideal structure for practical applications and pointed out aperiodic structured fibers, such as Fractal fibers, are a better option for low bend losses.
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For two perpendicular families of parallel lines this construction just gives the familiar square tiling of the plane, and for three families of lines at 120-degree angles from each other (themselves forming a trihexagonal tiling) this produces the rhombille tiling. However, for more families of lines this construction produces aperiodic tilings. In particular, for five families of lines at equal angles to each other (or, as de Bruijn calls this arrangement, a pentagrid) it produces a family of tilings that include the rhombic version of the Penrose tilings. The tetrakis square tiling is an infinite arrangement of lines forming a periodic tiling that resembles a multigrid with four parallel families, but in which two of the families are more widely spaced than the other two, and in which the arrangement is simplicial rather than simple.
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Determination of a material's surface reconstruction requires a measurement of the positions of the surface atoms that can be compared to a measurement of the bulk structure. While the bulk structure of crystalline materials can usually be determined by using a diffraction experiment to determine the Bragg peaks, any signal from a reconstructed surface is obscured due to the relatively tiny number of atoms involved. Special techniques are thus required to measure the positions of the surface atoms, and these generally fall into two categories: diffraction-based methods adapted for surface science, such as low-energy electron diffraction (LEED) or Rutherford backscattering spectroscopy, and atomic-scale probe techniques such as scanning tunneling microscopy (STM) or atomic force microscopy. Of these, STM has been most commonly used in recent history due to its very high resolution and ability to resolve aperiodic features.
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From the Introduction in Part I, Volume I, by Callimahos: "This text represents an extensive expansion and revision, both in scope and content, of the earlier work entitled 'Military Cryptanalysis, Part I' by William F. Friedman. This expansion and revision was necessitated by the considerable advancement made in the art since the publication of the previous text." Callimahos referred to parts III–VI at the end of the first volume: "...Part III will deal with varieties of aperiodic substitution systems, elementary cipher devices and cryptomechanisms, and will embrace a detailed treatment of cryptomathematics and diagnostic tests in cryptanalysis; Part IV will treat transposition and fractioning systems, and combined substitution-transposition systems; Part V will treat the reconstruction of codes, and the solution of enciphered code systems, and Part VI will treat the solution of representative machine cipher systems." However, parts IV–VI were never completed.
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A Penrose tiling In 1961, Hao Wang asked whether determining if a set of tiles admits a tiling of the plane is an algorithmically unsolvable problem or not. He conjectured that it is solvable, relying on the hypothesis that every set of tiles that can tile the plane can do it periodically (hence, it would suffice to try to tile bigger and bigger patterns until obtaining one that tiles periodically). Nevertheless, two years later, his student Robert Berger constructed a set of some 20,000 square tiles (now called "Wang tiles") that can tile the plane but not in a periodic fashion.A New Kind of Science As further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found. In 1976 Roger Penrose discovered a set of just two tiles, now referred to as Penrose tiles, that produced only non-periodic tilings of the plane.
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According to the NSA's BULLRUN Classification Guide, BULLRUN is not a Sensitive Compartmented Information (SCI) control system or compartment, but the codeword has to be shown in the classification line, after all other classification and dissemination markings. Furthermore, any details about specific cryptographic successes were recommend to be additionally restricted (besides being marked Top Secret//SI) with Exceptionally Controlled Information labels; a non- exclusive list of possible BULLRUN ECI labels was given as: APERIODIC, AMBULANT, AUNTIE, PAINTEDEAGLE, PAWLEYS, PITCHFORD, PENDLETON, PICARESQUE, and PIEDMONT without any details as to what these labels mean. Access to the program is limited to a group of top personnel at the Five Eyes (FVEY), the NSA and the signals intelligence agencies of the United Kingdom (GCHQ), Canada (CSE), Australia (ASD), and New Zealand (GCSB). Signals that cannot be decrypted with current technology may be retained indefinitely while the agencies continue to attempt to decrypt them.
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Accompanying the 11 year quasi-periodicity in sunspots, the large-scale dipolar (north-south) magnetic field component of the Sun also flips every 11 years, however, the peak in the dipolar field lags the peak in the sunspot number, with the former occurring at the minimum between two cycles. Levels of solar radiation and ejection of solar material, the number and size of sunspots, solar flares, and coronal loops all exhibit a synchronized fluctuation, from active to quiet to active again, with a period of 11 years. This cycle has been observed for centuries by changes in the Sun's appearance and by terrestrial phenomena such as auroras. Solar activity, driven both by the sunspot cycle and transient aperiodic processes govern the environment of the Solar System planets by creating space weather and impact space- and ground-based technologies as well as the Earth's atmosphere and also possibly climate fluctuations on scales of centuries and longer.
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Bindi has numerous international collaborations, especially with Princeton University, Harvard University, and the California Institute of Technology. His research activity, condensed in more than 300 scientific publications, has been devoted to four different areas: a) mantle mineralogy (clinopyroxenes, garnets, akimotoite, bridgmanite, hiroseite, ahrensite, wadsleyite, post- spinel phases, dense hydrous magnesium silicates); b) aperiodic structures in the mineral kingdom (melilite, fresnoite, calaverite, natrite, muthmannite, pearceite-polybasite, icosahedrite, decagonite); c) superstructures, twinning, OD-phenomena and structural complexity in minerals (melilites, pearceite, polybasite, samsonite, calaverite, empressite, fettelite, quadratite, sinnerite, sartorite, meneghinite, zinkenite); d) structure solution of unknown structures and description of new mineral species (about 250 crystal structures solved and ~100 new mineral species described) Significant among his research works are the crystal-chemical studies of major mineral phases for the Earth's mantle, and studies of potassium-rich clinopyroxene, which had broad international resonance. He is also very well known for his studies on the complexity of mineral structures integrating mineralogy with the most- advanced fields of crystallography.
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A sufficient but not necessary condition is detailed balance, which requires that each transition x \to x' is reversible: for every pair of states x, x', the probability of being in state x and transitioning to state x' must be equal to the probability of being in state x' and transitioning to state x, \pi(x) P(x' \mid x) = \pi(x') P(x \mid x'). # Uniqueness of stationary distribution: the stationary distribution \pi(x) must be unique. This is guaranteed by ergodicity of the Markov process, which requires that every state must (1) be aperiodic—the system does not return to the same state at fixed intervals; and (2) be positive recurrent—the expected number of steps for returning to the same state is finite. The Metropolis–Hastings algorithm involves designing a Markov process (by constructing transition probabilities) that fulfills the two above conditions, such that its stationary distribution \pi(x) is chosen to be P(x).
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Geometrical dissection of an L-tromino (rep-4) Both types of tromino can be dissected into n2 smaller trominos of the same type, for any integer n > 1\. That is, they are rep-tiles.. Continuing this dissection recursively leads to a tiling of the plane, which in many cases is an aperiodic tiling. In this context, the L-tromino is called a chair, and its tiling by recursive subdivision into four smaller L-trominos is called the chair tiling.. Motivated by the mutilated chessboard problem, Solomon W. Golomb used this tiling as the basis for what has become known as Golomb's tromino theorem: if any square is removed from a 2n × 2n chessboard, the remaining board can be completely covered with L-trominoes. To prove this by mathematical induction, partition the board into a quarter-board of size 2n−1 × 2n−1 that contains the removed square, and a large tromino formed by the other three quarter-boards.
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In higher symmetry groups and in reality, the vacuum is not a calm, randomly fluctuating, largely immaterial and passive substance, but at times can be viewed as a turbulent virtual plasma that can have complex vortices (i.e. solitons vis-à-vis particles), entangled states and a rich nonlinear structure. There are many observed nonlinear physical electromagnetic phenomena such as Aharonov–Bohm (AB) and Altshuler–Aronov–Spivak (AAS) effects, Berry, Aharonov–Anandan, Pancharatnam and Chiao–Wu phase rotation effects, Josephson effect, Quantum Hall effect, the de Haas–van Alphen effect, the Sagnac effect and many other physically observable phenomena which would indicate that the electromagnetic potential field has real physical meaning rather than being a mathematical artifact and therefore an all encompassing theory would not confine electromagnetism as a local force as is currently done, but as a SU(2) gauge theory or higher geometry. Higher symmetries allow for nonlinear, aperiodic behaviour which manifest as a variety of complex non-equilibrium phenomena that do not arise in the linearised U(1) theory, such as multiple stable states, symmetry breaking, chaos and emergence.
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A semigroup S that is a homomorphic image of a subsemigroup of T is said to be a divisor of T. The Krohn–Rhodes theorem for finite semigroups states that every finite semigroup S is a divisor of a finite alternating wreath product of finite simple groups, each a divisor of S, and finite aperiodic semigroups (which contain no nontrivial subgroups).Holcombe (1982) pp. 141–142 In the automata formulation, the Krohn–Rhodes theorem for finite automata states that given a finite automaton A with states Q and input set I, output alphabet U, then one can expand the states to Q' such that the new automaton A' embeds into a cascade of "simple", irreducible automata: In particular, A is emulated by a feed-forward cascade of (1) automata whose transitions semigroups are finite simple groups and (2) automata that are banks of flip-flops running in parallel.The flip-flop is the two-state automaton with three input operations: the identity (which leaves its state unchanged) and the two reset operations (which overwrite the current state by a resetting to a particular one of the two states).
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