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Clay Mathematics Institute

Clay Mathematics Institute
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NASA's Jet Propulsion Laboratory Blog Wang tile - Wikipedia This set of 11 Wang tiles will tile the plane but only aperiodically. Example of Wang tessellation with 13 tiles. The basic question about a set of Wang tiles is whether it can tile the plane or not, i.e., whether an entire infinite plane can be filled this way. Domino problem[edit] 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. The Domino Problem deals with the class of all domino sets. In other words, the domino problem asks whether there is an effective procedure that correctly settles the problem for all given domino sets. Aperiodic sets of tiles[edit] 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. Generalizations[edit] Applications[edit] In popular culture[edit] See also[edit]

Integrating Knowledge With Needs Chris Lucas "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."Albert Einstein (1879-1955) "Yet we also stress that truth is not the only aim of science.We want more than mere truth: what we look for is interesting truth...what we look for are answers to our problems." Introduction Scientists, in their attempts to maintain a detached 'objectivity' have always rejected the consideration of subjects, of values, of teleology, of purpose. To repair this long-running erroneous worldview we must first realise that science is about people - no people, no science. What Is It Like to be An Amoeba ? Rather than starting with humans, let us instead follow the path of evolution and start with that simple lifeform studied in high school biology. This purpose isn't a single dimensional one, even for such a simple animal. The Meaning of Knowledge Metascience as Integrator To do this we ask ourselves three simple questions:

Why are oxygen and hydrogen compressible, but water is barely compressible The force between two (non-reacting) atoms is approximately given by the Lennard-Jones potential, and this varies with the separation of the atoms something like this: (this image is from the Wikipedia article I linked above). In the diagram the parameter $\sigma$ can be thought of as the size of the atom, so the value on the $x$ axis of $r/\sigma = 1$ is the point where the atoms come into contact. Be cautious about taking this too iterally as atoms are somewhat fuzzy objects and don't have an exact size. nevertheless the point remains that there is a distance between the atoms at which they suddenly start to strongly repel each other. Now back to your question. Now conside water. You ask about compressing a mixture of (unreacted) oxygen and hydrogen.

Biodiversity is not just about saving exotic species from extinc Starting Monday, celebrations and events across the world will highlight the beginning of the UN's Year of International Biodiversity and the loss of our richly varied flaura and fauna, which is estimated to be as high as 1,000 times the natural rate as a result of human activities. Ahmed Djoghlaf, the general secretary of the treaty signed by 192 countries since 1992 to protect biodiversity, is blunt about efforts to preserve the health of biodiversity since the Rio Earth summit 18 years ago. Governments worldwide have failed to meet the treaty's target of reversing the trend for declining biodiversity, he says, and urgently need momentum to hit its targets for 2020. Biodiversity is integral to our daily lives. The equivalent to the Stern report for biodiversity is called The Economics of Ecosystems and Biodiversity (TEEB). Equally significant, are the vital natural services that the world's ecosystems provide.

Platonic Solid -- from Wolfram MathWorld The Platonic solids, also called the regular solids or regular polyhedra, are convex polyhedra with equivalent faces composed of congruent convex regular polygons. There are exactly five such solids (Steinhaus 1999, pp. 252-256): the cube, dodecahedron, icosahedron, octahedron, and tetrahedron, as was proved by Euclid in the last proposition of the Elements. The Platonic solids are sometimes also called "cosmic figures" (Cromwell 1997), although this term is sometimes used to refer collectively to both the Platonic solids and Kepler-Poinsot solids (Coxeter 1973). The Platonic solids were known to the ancient Greeks, and were described by Plato in his Timaeus ca. 350 BC. Schläfli (1852) proved that there are exactly six regular bodies with Platonic properties (i.e., regular polytopes) in four dimensions, three in five dimensions, and three in all higher dimensions. If 1. all lie on a sphere. 2. 3. 4. 5. Let (sometimes denoted ) be the number of polyhedron vertices, (or ) the number of faces.

Chinese meditation boosts brain activity U. OREGON (US)—Just 11 hours of learning a Chinese meditation technique boosts efficiency in a part of the brain that helps a person regulate behavior, according to new research. The technique—integrative body-mind training (IBMT)—has been the focus of scrutiny by researchers led by Yi-Yuan Tang of Dalian University of Technology in collaboration with University of Oregon psychologist Michael Posner. IBMT was adapted from traditional Chinese medicine in the 1990s in China, where it is practiced by thousands of people. The new research involves 45 students (28 males and 17 females); 22 subjects received IBMT while 23 participants were in a control group that received the same amount of relaxation training. Details are published online ahead of regular publication in the Proceedings of the National Academy of Sciences. A type of magnetic resonance called diffusion tensor imaging allowed researchers to examine fibers connecting brain regions before and after training.

Efficient Splitting of CO_2 in an Isothermal Redox Cycle Based on Ceria - CaltechAUTHORS Venstrom, Luke J. and De Smith, Robert M. and Hao, Yong and Haile, Sossina M. and Davidson, Jane H. (2014) Efficient Splitting of CO_2 in an Isothermal Redox Cycle Based on Ceria. Energy and Fuels, 28 (4). pp. 2732-2742. ISSN 1520-5029. Full text is not posted in this repository. Use this Persistent URL to link to this item: An isothermal thermochemical cycle to split CO_2 based on nonstoichiometric reduction and oxidation of ceria is demonstrated. Repository Staff Only: item control page Erowid