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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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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Each tiling in the sequence is indistinguishable from a true Ammann–Beenker tiling on a successively larger scale. Since some of these tilings are periodic, it follows that no decoration of the tiles which does force aperiodicity can be determined by looking at any finite patch of the tiling. The orientation of the vertex arrows which force aperiodicity, then, can only be deduced from the entire infinite tiling. The tiling has also an extremal property : among the tilings whose rhombuses alternate (that is, whenever two rhombuses are adjacent or separated by a row of square, they appear in different orientations), the proportion of squares is found to be minimal in the Ammann–Beenker tilings.
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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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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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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 solid is covalently networked and resistant to agglomeration and sintering. Beyond aperiodicity, these structures are used because the porous structure allows for rapid diffusion throughout the material, and the porous structure provides a large reaction surface. Fabrication is through coating the ambigel with a polymer electrolyte and then filling the void space with RuO2 colloids that act as an anode.
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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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Specifically, we can analyze different network structure to see for which structures these naive agents can successfully aggregate decentralized information. Since the DeGroot model can be considered a Markov chain, provided that a network is strongly connected (so there is a direct path from any agent to any other) and satisfies a weak aperiodicity condition, beliefs will converge to a consensus. When consensus is reached, the belief of each agent is a weighted average of agents' initial beliefs. These weights provide a measure of social influence.
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Markeas is part of the heritage of the Parisian spectral school, that of a "processual" music, a music more attached to sound than to the note, gradually moving from one state of the material to another; a music involving a dialectic between harmonic and inharmonic, between periodicity and aperiodicity. To this compositional approach, Alexandros Markeas adds a theatrical dimension, allowing him to escape from "pure music". His compositions are marked by the use of multimedia techniques. At some time he was a professor of generative improvisation at the Conservatoire de Paris (CNSMD).
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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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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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A vertex of a plane tiling or tessellation is a point where three or more tiles meet;M.V. Jaric, ed, Introduction to the Mathematics of Quasicrystals (Aperiodicity and Order, Vol 2) , Academic Press, 1989. generally, but not always, the tiles of a tessellation are polygons and the vertices of the tessellation are also vertices of its tiles. More generally, a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope; the vertices of other kinds of complexes such as simplicial complexes are its zero-dimensional faces.
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In 1962 he started using serial techniques, but in 1967 turned against the rhythmic aperiodicity and discontinuity characteristic of that technique . Instead, he began to use repetitive rhythmic structures similar to those of American minimalists such as Glass, Reich, Riley, and Young , though he developed these rhythmic ideas independently and retained the constructivism of serial thinking . He also continued to use twelve-tone rows often in his music, utilizing rows lacking thirds, perfect fifths, and semitones, in order to avoid suggestions of tonality and mutual attraction between pitches . His music is abstract, excluding emotion as either expression or goal .
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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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In 1967 Osamu Fujimura (MIT) showed basic advantages of the multi-band representation of speech ("An Approximation to Voice Aperiodicity", IEEE 1968). This work gave a start to development of the "multi-band excitation" method of speech coding, that was patented in 1997 (now expired) by founders of DVSI as "Multi-Band Excitation" (MBE). All consequent improvements known as Improved Multi-Band Excitation (IMBE), Advanced Multiband Excitation (AMBE), AMBE+ and AMBE+2 are based on this MBE method. AMBE is a codebook-based vocoder that operates at bitrates of between 2 and 9.6 kbit/s, and at a sampling rate of 8 kHz in 20-ms frames.
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Max Delbrück's thinking about the physical basis of life was an important influence on Schrödinger. However, long before the publication of What is Life?, geneticist and 1946 Nobel-prize winner H. J. Muller had in his 1922 article "Variation due to Change in the Individual Gene"American Naturalist 56 (1922) already laid out all the basic properties of the "heredity molecule" (then not yet known to be DNA) that Schrödinger was to re-derive in 1944 "from first principles" in What is Life? (including the "aperiodicity" of the molecule), properties which Muller specified and refined additionally in his 1929 article "The Gene As The Basis of Life"Proceedings of the International Congress of Plant Sciences 1 (1929) and during the 1930s.
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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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Observations of the luminosity of the star by the Kepler space telescope show small, frequent, non-periodic dips in brightness, along with two large recorded dips in brightness two years apart. The amplitude of the changes in the star's brightness, and the aperiodicity of the changes, mean that this star is of particular interest for astronomers. The star's changes in brightness are consistent with many small masses orbiting the star in "tight formation". The first major dip, on 5 March 2011, reduced the star's brightness by up to 15%, and the next 726 days later (on 28 February 2013) by up to 22%. (A third dimming, around 8%, occurred 48 days later.) In comparison, a planet the size of Jupiter would only obscure a star of this size by 1%, indicating that whatever is blocking light during the star's major dips is not a planet, but rather something covering up to half the width of the star.
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