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11,300 | The cognitive tutoring system developed by John Anderson at Carnegie Mellon University presents information from knowledge tracing as a skillometer, a visual graph of the learner's success in each of the monitored skills related to solving algebra problems. When a learner requests a hint, or an error is flagged, the kn... | Intelligent tutoring system | 0.818059 |
11,301 | Along with these, many other CAI projects were initiated in many countries including the US, the UK, and Canada.At the same time that CAI was gaining interest, Jaime Carbonell suggested that computers could act as a teacher rather than just a tool (Carbonell, 1970). A new perspective would emerge that focused on the us... | Intelligent tutoring system | 0.818059 |
11,302 | This conference was the first of its kind that was devoted to scientists and research in the field of AI. The latter part of the 1960s and 1970s saw many new CAI (Computer-Assisted instruction) projects that built on advances in computer science. The creation of the ALGOL programming language in 1958 enabled many schoo... | Intelligent tutoring system | 0.818059 |
11,303 | Turing's work as well as later projects by researchers such as Allen Newell, Clifford Shaw, and Herb Simon showed programs capable of creating logical proofs and theorems. Their program, The Logic Theorist exhibited complex symbol manipulation and even generation of new information without direct human control and is c... | Intelligent tutoring system | 0.818059 |
11,304 | A mass balance can also be taken differentially. The concept is the same as for a large mass balance, but it is performed in the context of a limiting system (for example, one can consider the limiting case in time or, more commonly, volume). A differential mass balance is used to generate differential equations that c... | Mass balance | 0.818058 |
11,305 | In physics, a mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique. The exact conse... | Mass balance | 0.818058 |
11,306 | These techniques are required for thorough design and analysis of systems such as the refrigeration cycle. In environmental monitoring, the term budget calculations is used to describe mass balance equations where they are used to evaluate the monitoring data (comparing input and output, etc.). In biology, the dynamic ... | Mass balance | 0.818058 |
11,307 | In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (and... | Separated set | 0.818054 |
11,308 | In software engineering, many reuse metrics and models are metrics used to measure code reuse and reusability. A metric is a quantitative indicator of an attribute of a thing. A model specifies relationships among metrics. | Reuse metrics | 0.818052 |
11,309 | He constructed an infinite family of non-Euclidean geometries by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space. The simplest of these is called elliptic geometry and it is considered a non-Euclidean geometry due to its lack of parallel lines.By formulating the geometry in terms... | Non-Euclidean space | 0.818042 |
11,310 | Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic... | Non-Euclidean space | 0.818042 |
11,311 | The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Circa 1813, Carl Friedrich Gauss and independently around 1818, the German professor of law Ferdinand Karl Schweikart had the germinal ideas of non-Euclidean geometry worked out, but neither published any r... | Non-Euclidean space | 0.818042 |
11,312 | These adjustments depend upon the axiom system used. Among others, these tweaks have the effect of modifying Euclid's second postulate from the statement that line segments can be extended indefinitely to the statement that lines are unbounded. Riemann's elliptic geometry emerges as the most natural geometry satisfying... | Non-Euclidean space | 0.818042 |
11,313 | The second case is not dealt with as easily. Simply replacing the parallel postulate with the statement, "In a plane, given a point P and a line l not passing through P, all the lines through P meet l", does not give a consistent set of axioms. This follows since parallel lines exist in absolute geometry, but this stat... | Non-Euclidean space | 0.818042 |
11,314 | As the first 28 propositions of Euclid (in The Elements) do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry.To obtain a non-Euclidean geometry, the parallel postulate (or its equivalent) must be replaced by its negation. Negating the Playfair... | Non-Euclidean space | 0.818042 |
11,315 | All approaches, however, have an axiom that is logically equivalent to Euclid's fifth postulate, the parallel postulate. Hilbert uses the Playfair axiom form, while Birkhoff, for instance, uses the axiom that says that, "There exists a pair of similar but not congruent triangles." In any of these systems, removal of th... | Non-Euclidean space | 0.818042 |
11,316 | Euclidean geometry can be axiomatically described in several ways. Unfortunately, Euclid's original system of five postulates (axioms) is not one of these, as his proofs relied on several unstated assumptions that should also have been taken as axioms. Hilbert's system consisting of 20 axioms most closely follows the a... | Non-Euclidean space | 0.818042 |
11,317 | The other two angles of a Saccheri quadrilateral are called the summit angles and they have equal measure. The summit angles of a Saccheri quadrilateral are acute if the geometry is hyperbolic, right angles if the geometry is Euclidean and obtuse angles if the geometry is elliptic. The sum of the measures of the angles... | Non-Euclidean space | 0.818042 |
11,318 | The fourth angle of a Lambert quadrilateral is acute if the geometry is hyperbolic, a right angle if the geometry is Euclidean or obtuse if the geometry is elliptic. Consequently, rectangles exist (a statement equivalent to the parallel postulate) only in Euclidean geometry. A Saccheri quadrilateral is a quadrilateral ... | Non-Euclidean space | 0.818042 |
11,319 | Euclidean and non-Euclidean geometries naturally have many similar properties, namely those that do not depend upon the nature of parallelism. This commonality is the subject of absolute geometry (also called neutral geometry). However, the properties that distinguish one geometry from others have historically received... | Non-Euclidean space | 0.818042 |
11,320 | Robert Heinlein's The Number of the Beast utilizes non-Euclidean geometry to explain instantaneous transport through space and time and between parallel and fictional universes. Zeno Rogue's HyperRogue is a roguelike game set on the hyperbolic plane, allowing the player to experience many properties of this geometry. M... | Non-Euclidean space | 0.818042 |
11,321 | The main character in Robert Pirsig's Zen and the Art of Motorcycle Maintenance mentioned Riemannian geometry on multiple occasions. In The Brothers Karamazov, Dostoevsky discusses non-Euclidean geometry through his character Ivan. Christopher Priest's novel Inverted World describes the struggle of living on a planet w... | Non-Euclidean space | 0.818042 |
11,322 | Non-Euclidean geometry is sometimes connected with the influence of the 20th-century horror fiction writer H. P. Lovecraft. In his works, many unnatural things follow their own unique laws of geometry: in Lovecraft's Cthulhu Mythos, the sunken city of R'lyeh is characterized by its non-Euclidean geometry. It is heavily... | Non-Euclidean space | 0.818042 |
11,323 | Non-Euclidean geometry often makes appearances in works of science fiction and fantasy. In 1895, H. G. Wells published the short story "The Remarkable Case of Davidson's Eyes". To appreciate this story one should know how antipodal points on a sphere are identified in a model of the elliptic plane. In the story, in the... | Non-Euclidean space | 0.818042 |
11,324 | In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative,... | Non-Euclidean space | 0.818042 |
11,325 | Australia Flinders University, Adelaide, South Australia Queensland University of Technology, Brisbane, Queensland The Australian National University, Canberra, Australian Capital Territory (a Bachelor of Mathematical Sciences BMASC) University of Adelaide, Adelaide, South Australia (a Bachelor of Mathematical Sciences... | Bachelor of Mathematics | 0.818041 |
11,326 | A Bachelor of Mathematics (abbreviated B.Math or BMath) is an undergraduate academic degree awarded for successfully completing a program of study in mathematics or related disciplines, such as applied mathematics, actuarial science, computational science, data analytics, financial mathematics, mathematical physics, pu... | Bachelor of Mathematics | 0.818041 |
11,327 | Waterhouse, William C. (1980). "The early proofs of Sylow's theorem". | Sylow p-subgroup | 0.818041 |
11,328 | Meo, M. (2004). "The mathematical life of Cauchy's group theorem". | Sylow p-subgroup | 0.818041 |
11,329 | An experiment in abstract algebra with Isabelle HOL" (PDF). J. Automat. Reason.. 23 (3): 235–264. | Sylow p-subgroup | 0.818041 |
11,330 | Kammüller, Florian; Paulson, Lawrence C. (1999). "A formal proof of Sylow's theorem. | Sylow p-subgroup | 0.818041 |
11,331 | Gow, Rod (1994). "Sylow's proof of Sylow's theorem". Irish Math. | Sylow p-subgroup | 0.818041 |
11,332 | Casadio, Giuseppina; Zappa, Guido (1990). "History of the Sylow theorem and its proofs". Boll. Storia Sci. | Sylow p-subgroup | 0.818041 |
11,333 | The Sylow theorems are a powerful statement about the structure of groups in general, but are also powerful in applications of finite group theory. This is because they give a method for using the prime decomposition of the cardinality of a finite group G {\displaystyle G} to give statements about the structure of its ... | Sylow p-subgroup | 0.818041 |
11,334 | Some non-prime numbers n are such that every group of order n is cyclic. One can show that n = 15 is such a number using the Sylow theorems: Let G be a group of order 15 = 3 · 5 and n3 be the number of Sylow 3-subgroups. Then n3 ∣ {\displaystyle \mid } 5 and n3 ≡ 1 (mod 3). The only value satisfying these constraints i... | Sylow p-subgroup | 0.818041 |
11,335 | The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group G {\displaystyle G} the order (number of elements) of every subgroup of G {\displaystyle G} divides the order of G {\displaystyle G} . The Sylow theorems state that for every prime factor p {\displays... | Sylow p-subgroup | 0.818041 |
11,336 | In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundament... | Sylow p-subgroup | 0.818041 |
11,337 | When (ei) is a partition of 1 in a Euclidean Jordan algebra E, the self-adjoint operators L(ei) commute and there is a decomposition into simultaneous eigenspaces. If a = Σ λi ei the eigenvalues of L(a) have the form Σ εi λi is 0, 1/2 or 1. The ei themselves give the eigenvalues λi. In particular an element a has non-n... | Structure group (Jordan algebra) | 0.818039 |
11,338 | The fundamental identity implies that c = Q(a)b is invertible if a and b are invertible and gives a formula for the inverse of Q(c). Applying it to c gives the inverse identity in full generality. Finally it can be verified immediately from the definitions that, if u = 1 − 2e for some idempotent e, then Q(u) is the per... | Structure group (Jordan algebra) | 0.818039 |
11,339 | Replacing a by its inverse, the relation follows if L(a) and L(a−1) are invertible. If not it holds for a + ε1 with ε arbitrarily small and hence also in the limit. These identities are easy to prove in a finite-dimensional (Euclidean) Jordan algebra (see below) or in a special Jordan algebra, i.e. the Jordan algebra d... | Structure group (Jordan algebra) | 0.818039 |
11,340 | But then SL(2,k)/K ≅ U/U ∩ K. The right hand side here is Abelian while the left hand side is its own commutator subgroup. Hence this must be the trivial group and K = SL(2,k). Given an element a in the complex Jordan algebra A = EC, the unital Jordan subalgebra C is associative and commutative. | Structure group (Jordan algebra) | 0.818039 |
11,341 | In this way the classification of Euclidean Jordan algebras is reduced to that of simple ones. For a simple algebra E all inner products for which the operators L(a) are self adjoint are proportional. Indeed, any other product has the form (Ta, b) for some positive self-adjoint operator commuting with the L(a)'s. Any n... | Structure group (Jordan algebra) | 0.818039 |
11,342 | Minimality forces Ei to be simple, i.e. to have no non-trivial ideals. For since L(ei) commutes with all L(a)'s, any ideal F ⊂ Ei would be invariant under E since F = eiF. Such a decomposition into a direct sum of simple Euclidean algebras is unique. | Structure group (Jordan algebra) | 0.818039 |
11,343 | The idempotent e is central in E, where the center of E is defined to be the set of all z such that L(z) commutes with L(a) for all a. It forms a commutative associative subalgebra. Continuing in this way E can be written as a direct sum of minimal ideals E = ⊕ E i . | Structure group (Jordan algebra) | 0.818039 |
11,344 | The set S of invertible elements u such that u* = u−1 can be characterized equivalently either as those u for which L(u) is a normal operator with uu* = 1 or as those u of the form exp ia for some a in E. In particular S is connected. The identity component of Γu(EC) acts transitively on S g in GL(EC) is in the unitary... | Structure group (Jordan algebra) | 0.818039 |
11,345 | {\displaystyle \displaystyle {(,c)=(b,).}} Combining these expressions gives ( , c ) = 0 , {\displaystyle \displaystyle {(,c)=0,}} which implies the Jordan identity. Finally the positive cone of E coincides with C. This depends on the fact that in any Euclidean Jordan algebra E Q ( e a ) = e 2 L ( a ) . | Structure group (Jordan algebra) | 0.818039 |
11,346 | Thus K is the fixed point subgroup of σ. Let g {\displaystyle {\mathfrak {g}}} be the Lie algebra of G. Thus σ induces an involution of g {\displaystyle {\mathfrak {g}}} and hence a ±1 eigenspace decomposition g = k ⊕ p , {\displaystyle \displaystyle {{\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}},}} where k {\d... | Structure group (Jordan algebra) | 0.818039 |
11,347 | They are valid in any Jordan algebra. This was conjectured by Jacobson and proved in Macdonald (1960): Macdonald showed that if a polynomial identity in three variables, linear in the third, is valid in any special Jordan algebra, then it holds in all Jordan algebras.In fact for c in A and F(a) a function on A with val... | Structure group (Jordan algebra) | 0.818039 |
11,348 | Taking adjoints it follows that L(a) commutes with L(e). Similarly if (1 − e)a = 0, L(a) commutes with I − L(e) and hence L(e). This implies the Jordan algebra and module properties. To check that a product of elements in the module lies in the algebra, it is enough to check this for squares: but if L(e)a = 1/2 a, then... | Structure group (Jordan algebra) | 0.818039 |
11,349 | The subspace E1/2(e) is a module for the centralizer of e, the centralizer module, and the product of any two elements in it lies in the centralizer subalgebra. On the other hand, if U = 8 L ( e ) 2 − 8 L ( e ) + I , {\displaystyle \displaystyle {U=8L(e)^{2}-8L(e)+I,}} then U is self-adjoint equal to 1 on the centraliz... | Structure group (Jordan algebra) | 0.818039 |
11,350 | There is a corresponding orthogonal eigenspace decomposition of E E = E 0 ( e ) ⊕ E 1 / 2 ( e ) ⊕ E 1 ( e ) , {\displaystyle \displaystyle {E=E_{0}(e)\oplus E_{1/2}(e)\oplus E_{1}(e),}} where, for a in E, Eλ(a) denotes the λ-eigenspace of L(a). In this decomposition E1(e) and E0(e) are Jordan algebras with identity ele... | Structure group (Jordan algebra) | 0.818039 |
11,351 | If E is a Euclidean Jordan algebra an ideal F in E is a linear subspace closed under multiplication by elements of E, i.e. F is invariant under the operators L(a) for a in E. If P is the orthogonal projection onto F it commutes with the operators L(a), In particular F⊥ = (I − P)E is also an ideal and E = F ⊕ F⊥. Furthe... | Structure group (Jordan algebra) | 0.818039 |
11,352 | {\displaystyle \displaystyle {G=KAN.}} For i ≠ j let g i j = { X ∈ g: = 1 2 ( α i − α j ) X , f o r a = ∑ α i e i } . {\displaystyle \displaystyle {{\mathfrak {g}}_{ij}=\{X\in {\mathfrak {g}}:={1 \over 2}(\alpha _{i}-\alpha _{j})X,\,\,\,\mathrm {for} \,\,\,a=\sum \alpha _{i}e_{i}\}.}} Then the Lie algebra of N is n = ... | Structure group (Jordan algebra) | 0.818039 |
11,353 | Since the group GT has trivial center, the map into GL( g T {\displaystyle {\mathfrak {g}}_{T}} ) is injective. Since KT is compact, its image in GL( g T {\displaystyle {\mathfrak {g}}_{T}} ) is compact. Since the Lie algebra g T {\displaystyle {\mathfrak {g}}_{T}} is compatible with that of ANT, the image of ANT is cl... | Structure group (Jordan algebra) | 0.818039 |
11,354 | As a group HT normalizes g T {\displaystyle {\mathfrak {g}}_{T}} since the two subgroups KT and ANT do. The +1 eigenspace corresponds to the Lie algebra of KT. Similarly the Lie algebras of the linear group AN and the affine group N0 lie in g T {\displaystyle {\mathfrak {g}}_{T}} . | Structure group (Jordan algebra) | 0.818039 |
11,355 | By a result of Henri Cartan, HD is a Lie group. Cartan's original proof is presented in Narasimhan (1971). It can also be deduced from the fact the D is complete for the Bergman metric, for which the isometries form a Lie group; by Montel's theorem, the group of biholomorphisms is a closed subgroup.That HT is a Lie gro... | Structure group (Jordan algebra) | 0.818039 |
11,356 | An idempotent g is called primitive or minimal if it cannot be written as a sum of non-zero orthogonal idempotents. If e1, ..., em are pairwise orthogonal idempotents then their sum is also an idempotent and the algebra they generate consists of all linear combinations of the ei. It is an associative algebra. | Structure group (Jordan algebra) | 0.818039 |
11,357 | Jordan frames provide one of the main Jordan algebraic techniques to describe the symmetry groups. Each Jordan frame gives rise to a product of copies of R and C. The symmetry groups of the corresponding open domains and the compactification—polydisks and polyspheres—can be deduced from the case of the unit disk, the u... | Structure group (Jordan algebra) | 0.818039 |
11,358 | All these features generalize to arbitrary Euclidean Jordan algebras. The compactification and complex Lie group are described in the next section and correspond to the dual Hermitian symmetric space of compact type. In this section only the symmetries of and between the bounded domain and tube domain are described. | Structure group (Jordan algebra) | 0.818039 |
11,359 | The groups are also algebraic. They have distinguished generating subgroups and have an explicit description in terms of generators and relations. Moreover, the Cayley transform gives an explicit Möbius transformation from the open disk onto the upper half plane. | Structure group (Jordan algebra) | 0.818039 |
11,360 | According to the "elementary approach" to bounded symmetric space of Koecher, Hermitian symmetric spaces of noncompact type can be realized in the complexification of a Euclidean Jordan algebra E as either the open unit ball for the spectral norm, a bounded domain, or as the open tube domain T = E + iC, where C is the ... | Structure group (Jordan algebra) | 0.818039 |
11,361 | Every g in Γ(EC) has the form g = h Q(a) with h an automorphism and a invertible.The unitary structure group Γu(EC) is the subgroup of Γ(EC) consisting of unitary operators, so that Γu(EC) = Γ(EC) ∩ U(EC). The stabilizer of 1 in Γu(EC) is Aut E. Every g in Γu(EC) has the form g = h Q(u) with h in Aut E and u invertible... | Structure group (Jordan algebra) | 0.818039 |
11,362 | If g is in the structure group and a is invertible, ga is also invertible with (ga)−1 = (gt)−1a−1. If E is simple, Γ(E) = Aut C × {±1}, Γ(E) ∩ O(E) = Aut E × {±1} and the identity component of Γ(E) acts transitively on C. Γ(EC) is the complexification of Γ(E), which has Lie algebra k ⊕ p {\displaystyle {\mathfrak {k}}\... | Structure group (Jordan algebra) | 0.818039 |
11,363 | Let V be a finite dimensional real inner product space and set E = V ⊕ R with inner product (u⊕λ,v⊕μ) =(u,v) + λμ and product (u⊕λ)∘(v⊕μ)=( μu + λv) ⊕ . This is a Euclidean Jordan algebra of rank 2, called a spin factor. The above examples in fact give all the simple Euclidean Jordan algebras, except for one exceptiona... | Structure group (Jordan algebra) | 0.818039 |
11,364 | Let Hn(H) be the space of self-adjoint n by n matrices with entries in the quaternions, inner product (a,b) = Re Tr ab* and Jordan product a ∘ b = 1/2(ab + ba). Then Hn(H) is a simple Euclidean Jordan algebra of rank n for n ≥ 3. | Structure group (Jordan algebra) | 0.818039 |
11,365 | Let Hn(R) be the space of real symmetric n by n matrices with inner product (a,b) = Tr ab and Jordan product a ∘ b = 1/2(ab + ba). Then Hn(R) is a simple Euclidean Jordan algebra of rank n for n ≥ 3. Let Hn(C) be the space of complex self-adjoint n by n matrices with inner product (a,b) = Re Tr ab* and Jordan product a... | Structure group (Jordan algebra) | 0.818039 |
11,366 | The trace form has the advantage of being manifestly invariant under automorphisms of the Jordan algebra, which is thus a closed subgroup of O(E) and thus a compact Lie group. In practical examples, however, it is often easier to produce an inner product for which the L(a) are self-adjoint than verify directly positive... | Structure group (Jordan algebra) | 0.818039 |
11,367 | Let C be the positive cone in a simple Euclidean Jordan algebra E. Aut C is the closed subgroup of GL(E) taking C (or its closure) onto itself. Let G = Aut0 C be the identity component of Aut C and let K be the closed subgroup of G fixing 1. From the group theoretic properties of cones, K is a connected compact subgrou... | Structure group (Jordan algebra) | 0.818039 |
11,368 | In the context of a triangle, barycentric coordinates are also known as area coordinates or areal coordinates, because the coordinates of P with respect to triangle ABC are equivalent to the (signed) ratios of the areas of PBC, PCA and PAB to the area of the reference triangle ABC. Areal and trilinear coordinates are u... | Barycentric coordinate system (mathematics) | 0.818037 |
11,369 | Molecular Biology Reports is a monthly peer-reviewed scientific journal covering research on normal and pathological molecular processes. | Molecular Biology Reports | 0.818036 |
11,370 | Human genetics is the study of inheritance as it occurs in human beings. Human genetics encompasses a variety of overlapping fields including: classical genetics, cytogenetics, molecular genetics, biochemical genetics, genomics, population genetics, developmental genetics, clinical genetics, and genetic counseling. Gen... | Human genetics | 0.81803 |
11,371 | A karyotype is a very useful tool in cytogenetics. A karyotype is picture of all the chromosomes in the metaphase stage arranged according to length and centromere position. A karyotype can also be useful in clinical genetics, due to its ability to diagnose genetic disorders. On a normal karyotype, aneuploidy can be de... | Human genetics | 0.81803 |
11,372 | In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation ... | Multiplication tables | 0.818029 |
11,373 | In abstract algebra, one uses homology to define derived functors, for example the Tor functors. Here one starts with some covariant additive functor F and some module X. The chain complex for X is defined as follows: first find a free module F 1 {\displaystyle F_{1}} and a surjective homomorphism p 1: F 1 → X . {\disp... | Homology theory | 0.818025 |
11,374 | It is a theorem that the re-glued surface must self-intersect (when immersed in Euclidean 3-space). Like the torus, cycles a and b cannot be shrunk while c can be. But unlike the torus, following b forwards right round and back reverses left and right, because b happens to cross over the twist given to one join. | Homology theory | 0.818025 |
11,375 | On the ordinary sphere S 2 {\displaystyle S^{2}} , the cycle b in the diagram can be shrunk to the pole, and even the equatorial great circle a can be shrunk in the same way. The Jordan curve theorem shows that any arbitrary cycle such as c can be similarly shrunk to a point. All cycles on the sphere can therefore be c... | Homology theory | 0.818025 |
11,376 | All three implement pre-processing algorithms based on simple-homotopy equivalence and discrete Morse theory to perform homology-preserving reductions of the input cell complexes before resorting to matrix algebra. Kenzo is written in Lisp, and in addition to homology it may also be used to generate presentations of ho... | Homology theory | 0.818025 |
11,377 | Chapter twelve, titled classical radiation-theory is completely new and focuses on the empirical development of spectral series as well as the historical development of black body radiation physics. The final chapter, chapter eight, was renamed to classical theory in the age of Lorentz and contains new material, while ... | A History of the Theories of Aether and Electricity | 0.818022 |
11,378 | Chapter one of the first volume was renamed the theory of the aether to the death of Newton after being mostly rewritten, though it still focuses on René Descartes, Isaac Newton, Pierre de Fermat, Robert Hooke, and Christiaan Huygens, among others. The chapter begins with a discussion of physics from the initial formul... | A History of the Theories of Aether and Electricity | 0.818022 |
11,379 | The second volume, subtitled The Modern Theories (1900–1926), was originally published in 1951 by Thomas Nelson and Sons. The book is the continuation of Whittaker's survey of the history of physics into the period 1900–1926 and describes the revolution in physics over the first quarter of the 20th century. The major h... | A History of the Theories of Aether and Electricity | 0.818022 |
11,380 | The first volume, subtitled The Classical Theories, was initially published in 1951 by Thomas Nelson and Sons. The book is a revision of the original 1910 book, with an added chapter on classical radiation theory, some new material, but remains focused on pre-1900 physics. The book has a similar scope as the first edit... | A History of the Theories of Aether and Electricity | 0.818022 |
11,381 | The final two chapters give an account of the birth of quantum mechanics. Matrix mechanics is discussed in chapter eight, including the Heisenberg picture and the introduction of physical operators. Erwin Schrödinger, the Schrödinger picture, and Schrödinger's equation are all discussed in the final chapter. | A History of the Theories of Aether and Electricity | 0.818022 |
11,382 | Chapter four, on spectroscopy in old quantum theory, discusses many of Niels Bohr's precursors, including Arthur W. Conway, Penry Vaughan Bevan, John William Nicholson, and Niels Bjerrum. Chapter five switches to gravitation, discussing the history of cosmology and the general theory of relativity. Chapter six returns ... | A History of the Theories of Aether and Electricity | 0.818022 |
11,383 | The first chapter, the age of Rutherford, discusses the state of empirical physics at the turn of the twentieth century. Chapter two discusses is on the origins of special relativity and is highly controversial, and is the base of Whittaker's role in the relativity priority dispute. In this chapter, as the title sugges... | A History of the Theories of Aether and Electricity | 0.818022 |
11,384 | Born also points out that the book goes beyond what ordinary textbooks can do, which he believes offer students "the shortest and simplest way to knowledge and understanding", and "are in cases not only unhistorical but a distortion of history".Freeman Dyson, in his 1954 review, remarks that the second volume has, by n... | A History of the Theories of Aether and Electricity | 0.818022 |
11,385 | And points out that the last two chapters of the book give a "detailed and lively account of the birth of quantum mechanics in both of its forms, matrix mechanics and wave mechanics." He also praises Whittaker for setting aside his philosophical interests, saying "Whittaker the conscientious historian of science, has t... | A History of the Theories of Aether and Electricity | 0.818022 |
11,386 | In a September 1953 letter to Albert Einstein published in 1971, Max Born writes that, other than the relativity priority issues, it was "particularly unpleasant" for him that Whittaker "had woven all sorts of personal information into his account of quantum mechanics" while Born's role in the development was "extolled... | A History of the Theories of Aether and Electricity | 0.818022 |
11,387 | Edmond Whittaker under the same title in 1910. It is, rather, a thorough and authoritative chronicle of the development of theoretical physics from in the period 1900–1926, including atomic structure, special relativity, quantum theory, general relativity, matrix mechanics, and wave mechanics".A review by P. W. Bridgm... | A History of the Theories of Aether and Electricity | 0.818022 |
11,388 | In his 1956 book Physics in My Generation, Born goes on to call it an "excellent book" and talks about using the first edition as a reference when he was a student.Freeman Dyson, in a 1954 review, said the second volume is "more limited and professional in its scope" than the first volume, giving a "clear, logical acco... | A History of the Theories of Aether and Electricity | 0.818022 |
11,389 | In a 1954 book review of the second volume, Max Born praised both volumes of the expanded and revised second edition, saying "his second volume is a magnificent work, excellent not only through a brilliant style and clarity of expression, but also through an incredible scholarship and erudition" and that "this work mak... | A History of the Theories of Aether and Electricity | 0.818022 |
11,390 | The books are considered authoritative texts on the developments of classical electromagnetism and continue to be cited by widely adopted textbooks on the subject. A third volume, covering the years 1925 to 1950, was promised in the second edition but was never published, as Whittaker died in 1956. The two volumes prov... | A History of the Theories of Aether and Electricity | 0.818022 |
11,391 | Arthur Tyndall, in his 1951 review, stated the book is "rich in experimental fact", with comparatively fewer mathematical sections, with notable exceptions such as those on Lorentz and Maxwell, saying that "this new volume is not a heavy treatise in theoretical physics, as perhaps its name might suggest". William McCre... | A History of the Theories of Aether and Electricity | 0.818022 |
11,392 | Miller noted that while it is primarily a history book, it is also "philosophy, physics, and mathematics of the first temper" and that it gives an "elegant penetrating examination of The Classical Theories". He also noted that although it is "heavy reading", the work is "delightfully clear" and that the "documentation ... | A History of the Theories of Aether and Electricity | 0.818022 |
11,393 | Whittaker believed that a new edition should include the developments in physics that took part at the turn of the twentieth century and declined to have it reprinted. He wrote the second edition of the book after his retirement and published The Classical Theories in 1951, which also received critical acclaim. In the ... | A History of the Theories of Aether and Electricity | 0.818022 |
11,394 | The first volume, subtitled The Classical Theories, was published in 1951 and served as a revised and updated edition to the first book. The second volume, subtitled The Modern Theories (1900–1926), was published two years later in 1953, extended this work covering the years 1900 to 1926. Notwithstanding a notorious co... | A History of the Theories of Aether and Electricity | 0.818022 |
11,395 | He also continued performing fundamental research in analytical dynamics at Trinity College in Dublin throughout this period.The original version of the book was universally praised and was considered an authoritative reference work in the history of physics, despite its difficulty to obtain past the 1920s. When the fi... | A History of the Theories of Aether and Electricity | 0.818022 |
11,396 | "Whittaker wrote the first edition in his spare time while he was thirty-seven years old, during which time he was serving as Royal Astronomer of Ireland from 1906 onwards. The post's relative ease allowed him to devote time to reading for the project, which he worked on until its release in 1910. During this same peri... | A History of the Theories of Aether and Electricity | 0.818022 |
11,397 | As a teacher at Trinity College, beginning after his election as a fellow in 1896, Whittaker gave advanced lectures in spectroscopy, astrophysics, and electro-optics. His first book, Modern Analysis, was initially published in 1902 and remained a standard reference for applied mathematicians. His second major release, ... | A History of the Theories of Aether and Electricity | 0.818022 |
11,398 | The book was originally written in the period immediately following the publication of Einstein's Annus Mirabilis papers and several years following the early work of Max Planck; it was a transitional period for physics, where special relativity and old quantum theory were gaining traction. The book serves to document ... | A History of the Theories of Aether and Electricity | 0.818022 |
11,399 | Torretti notes that Poincare's failure to catch on was his notorious conventionalism, and the fact that he may have been a little too proud to admit that "he had lost the glory of founding 20th-century physics to a young Swiss patent clerk. "Charles Scribner, in his 1984 article Henri Poincaré and the Principle of Rela... | A History of the Theories of Aether and Electricity | 0.818022 |
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