Einstein notation - Wikipedia Shorthand notation for tensor operations In mathematics, especially in applications of linear algebra to physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. Introduction[edit] Statement of convention[edit] is simplified by the convention to: The upper indices are not exponents but are indices of coordinates, coefficients or basis vectors. In general relativity, a common convention is that In general, indices can range over any indexing set, including an infinite set. An index that is summed over is a summation index, in this case "i ". An index that is not summed over is a free index and should appear only once per term. Application[edit] for the
Discovery of classic pi formula a ‘cunning piece of magic’ : NewsCenter While most people associate the mathematical constant π (pi) with arcs and circles, mathematicians are accustomed to seeing it in a variety of fields. But two University scientists were still surprised to find it lurking in a quantum mechanics formula for the energy states of the hydrogen atom. “We didn’t just find pi,” said Tamar Friedmann, a visiting assistant professor of mathematics and a research associate of high energy physics, and co-author of a paper published this week in the Journal of Mathematical Physics. “We found the classic seventeenth century Wallis formula for pi, making us the first to derive it from physics, in general, and quantum mechanics, in particular.” The Wallis formula—developed by British mathematician John Wallis in his book Arithmetica Infinitorum—defines π as the product of an infinite string of ratios made up of integers. Friedmann did not set out to look for π nor for the Wallis formula. which can be reduced to the classic Wallis formula.
En el Día de Pi, 10 curiosidades sobre el número irracional El matemático griego Arquímedes fue uno de los primeros en aproximar su valor. Para aquellos que deseen conocer cuánto mide, solo deberán calcular el perímetro de una circunferencia y dividirlo por su diámetro. Blogthinkbig.com | Cada 14 de marzo se celebra en todo el mundo el Día de Pi, una conmemoración que trata de promover la divulgación sobre el número π en particular y acercar las matemáticas a la sociedad. Por eso nos unimos a esta fiesta matemática para fomentar la difusión científica. ¿Qué es el número Pi? El número Pi se define como la relación que existe entre la longitud o el perímetro de una circunferencia y su diámetro. ¿Cómo se calcula Pi? El matemático griego Arquímedes fue uno de los primeros en aproximar el valor del número Pi. ¿Pi vale 3,14? No. ¿Qué significa que sea un número irracional? π suele ser descrito como un número irracional y trascendente. ¿Podemos saber todos sus decimales? ¿Para qué sirve? ¿Es tan importante como parece? ¿Por qué se celebra el Día de Pi?
The Most Mathematically Perfect Day of the Year - Scientific American Blog Network Whether you write it 6/28 or 28/6, today is a perfect day. A perfect number is a number that is the sum of its factors besides itself, and 6 (1+2+3) and 28 (1+2+4+7+14) are the first two perfect numbers. Hence, June 28 is a perfect day. Mathematical date enthusiasts will note that today is also Tau Day, if you didn’t get enough circle constant celebration in March. If you’d prefer to keep the celebration mathematical, you could of course go looking for the next perfect number. No matter how you choose to celebrate, please avoid any involvement in perfect crimes or perfect storms.
The Tao of Tau - Scientific American Blog Network “It is lamentable that there’s no famous dessert named ‘tau,’” Michael Hartl told me recently at a sunny, stylish café in Venice, California. He reluctantly admitted that pi, the constant approximately equal to 3.14, has this one advantage over tau, a number he introduced to replace it. Pastry puns aside, Hartl has achieved minor internet fame for arguing that tau is superior to its vastly better known cousin. In his popular 2010 “Tau Manifesto,” inspired by Bob Palais’ 2001 essay “Pi Is Wrong,” Hartl posits that pi, the ratio of a circle’s circumference to its diameter, creates unnecessary complications in many formulas. A more appropriate number to work with when it comes to circles would be 2pi, or about 6.28. He named that number tau, and declared June 28 (6/28) to be Tau Day. “The circle constant ought to be defined in terms of radius,” Hartl told me over the chatter of other café patrons. Tau Protein Interest has soared in exploring tau’s role in these diseases. Tau Lepton Other Uses
El genio babilonio que se adelantó a Pitágoras 1.000 años y creó las tablas de trigonometría más precisas del mundo - BBC Mundo Derechos de autor de la imagen UNSW/Andrew Kelly En todo triángulo rectángulo el cuadrado de la hipotenusa es igual a la suma de los cuadrados de los catetos.... ¿te acuerdas o preferiste olvidarlo? ¿Qué me dices de los senos y cosenos; tangentes y cotangentes; secantes y cosecantes? ¿Qué dirías si te contaran que al menos 1.000 años antes de que el matemático griego Pitágoras (569-475 a.C.) se pusiera a pensar en triángulos y de que su compatriota Hiparco de Nicea (190-120 a.C.) se inventara la trigonometría los babilonios sabían hacer lo mismo pero de una manera menos complicada y más precisa que la que heredamos de los griegos y seguimos usando hoy? Pues precisamente eso fue lo que revelaron Daniel Mansfield y Norman Wildberger, de la Escuela de Matemáticas y Estadística de la Facultad de Ciencias de la Universidad de Nueva Gales del Sur, Australia. La misteriosa Plimpton 322 Trigonometría viene del griego 'trigonon', que significa triángulo, y metron, medida" iSTOCK Y tenían otra ventaja.
Levi-Civita symbol Antisymmetric permutation object acting on tensors In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, ..., n, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ, or less commonly the Latin lower case e. where each index i1, i2, ..., in takes values 1, 2, ..., n. If any two indices are equal, the symbol is zero. where p (called the parity of the permutation) is the number of pairwise interchanges of indices necessary to unscramble i1, i2, ..., in into the order 1, 2, ..., n, and the factor (−1)p is called the sign, or signature of the permutation. or Minkowski space. Definition[edit] Two dimensions[edit] Three dimensions[edit] Some examples: Product[edit] is
Ricci calculus Tensor index notation for tensor-based calculations A component of a tensor is a real number that is used as a coefficient of a basis element for the tensor space. The tensor is the sum of its components multiplied by their corresponding basis elements. For compactness and convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal quantification over free indices. Notation for indices[edit] Basis-related distinctions[edit] Space and time coordinates[edit] Where a distinction is to be made between the space-like basis elements and a time-like element in the four-dimensional spacetime of classical physics, this is conventionally done through indices as follows:[6] Some sources use 4 instead of 0 as the index value corresponding to time; in this article, 0 is used. Coordinate and index notation[edit] The author(s) will usually make it clear whether a subscript is intended as an index or as a label. where See also[edit]
Math Mystery: Shinichi Mochizuki and the Impenetrable Proof Sometime on the morning of August 30 2012, Shinichi Mochizuki quietly posted four papers on his website. The papers were huge—more than 500 pages in all—packed densely with symbols, and the culmination of more than a decade of solitary work. They also had the potential to be an academic bombshell. In them, Mochizuki claimed to have solved the abc conjecture, a 27-year-old problem in number theory that no other mathematician had even come close to solving. If his proof was correct, it would be one of the most astounding achievements of mathematics this century and would completely revolutionize the study of equations with whole numbers. Mochizuki, however, did not make a fuss about his proof. Probably the first person to notice the papers was Akio Tamagawa, a colleague of Mochizuki's at RIMS. Fesenko e-mailed some top experts in Mochizuki's field of arithmetic geometry, and word of the proof quickly spread. Adding to the enigma is Mochizuki himself. And that, says Faltings, is a problem.
The Golden Ratio: Design's Biggest Myth In the world of art, architecture, and design, the golden ratio has earned a tremendous reputation. Greats like Le Corbusier and Salvador Dalí have used the number in their work. The Parthenon, the Pyramids at Giza, the paintings of Michelangelo, the Mona Lisa, even the Apple logo are all said to incorporate it. It's bullshit. What Is The Golden Ratio? First described in Euclid's Elements 2,300 years ago, the established definition is this: two objects are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. In plain English: if you have two objects (or a single object that can be split into two objects, like the golden rectangle), and if, after you do the math above, you get the number 1.6180, it's usually accepted that those two objects fall within the golden ratio. The Golden Ratio as Mozart Effect It's pedantic, sure. But there isn't. One guy who believed this was Adolf Zeising. He was a long-winded guy.
Los misteriosos ‘círculos de hadas’ confirman las teorías de Alan Turing Un grupo de investigadores ha descubierto en el desierto occidental de Australia unos misteriosos claros entre la vegetación. Vistos desde arriba, enseguida llaman la atención dos cosas: por un lado, la forma circular de las calvas y, por el otro, el patrón hexagonal que forman los círculos entre sí. El fenómeno no es nuevo, los más famosos se encontraron en Namibia (África). Los círculos de hadas de Namibia (también los hay en las vecinas Angola y Sudáfrica) llevan décadas intrigando a los científicos. Ahora, un grupo de científicos cuestiona el papel de los insectos o las hierbas tóxicas. Los círculos de los desiertos de Namibia y Australia siguen un patrón similar al de las células de la piel En los alrededores del poblado minero de Newman, en la región australiana de Pilbara, han hallado grandes extensiones de vegetación de matorral salpicadas de círculos similares a los de Namibia. Esta autoorganización es lo que planteó Turing en uno de sus últimos trabajos.
Fractal Dimension Index o índice de dimensión fractal • esBolsa El Fractal Dimension Index (o índice de dimensión fractal) se basa en la idea de que los mercados financieros son parte de un proceso natural y no mecánico, y por tanto son caóticos. Es por eso por lo que deben usarse herramientas no lineales para predecir la dinámica de los mercados, y aquí es donde entra la geometría fractal. Todos los sistemas caóticos tienen una medida cuantitativa que se conoce como “dimensión fractal”. La dimensión fractal describe cómo el objeto ocupa su lugar en el espacio. Un fractal es un objeto en el que sus partes individuales son similares al conjunto. Por eso, los fractales mantienen sus dimensiones independientemente de la escala que se utilice. Historia La historia del Fractal Dimension Index comienza con el hidrólogo y constructor de diques británico H.E. Para resolver el problema, Hurst consideró la relación entre las lluvias a lo largo del año, los extremos de máxima y mínima agua y el nivel de la presa. Interpretación del Fractal Dimension Index