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87,400 | Quickly the process became an industry standard to allow weight distribution around the perimeter of the clubhead. Other industries that use standard investment-cast parts include military, aerospace, medical, jewelry, airline, automotive and golf clubs especially since the start of 3D printing technology. With the inc... | Investment Casting | 0.78209 |
87,401 | He has been a professor in the department of mathematics of the University of Illinois at Urbana-Champaign since 2001. Prior to this appointment, he was a faculty member at the University of South Carolina. Ford received a Bachelor of Science in Computer Science and Mathematics in 1990 from the California State Univers... | Kevin Ford (mathematician) | 0.78209 |
87,402 | Kevin B. Ford (born 22 December 1967) is an American mathematician working in analytic number theory. | Kevin Ford (mathematician) | 0.78209 |
87,403 | Applications of regular tree grammars include: Instruction selection in compiler code generation A decision procedure for the first-order logic theory of formulas over equality (=) and set membership (∈) as the only predicates Solving constraints about mathematical sets The set of all truths expressible in first-order ... | Regular tree grammar | 0.782089 |
87,404 | A regular tree grammar G is defined by the tuple G = (N, Σ, Z, P), where N is a finite set of nonterminals, Σ is a ranked alphabet (i.e., an alphabet whose symbols have an associated arity) disjoint from N, Z is the starting nonterminal, with Z ∈ N, and P is a finite set of productions of the form A → t, with A ∈ N, an... | Regular tree grammar | 0.782089 |
87,405 | The grammar G1 corresponds to the algebraic data type declarations (in the Standard ML programming language): Every member of L(G1) corresponds to a Standard-ML value of type BList. For another example, let G2 = (N1, Σ1, BList1, P1 ∪ P2), using the nonterminal set and the alphabet from above, but extending the producti... | Regular tree grammar | 0.782089 |
87,406 | In theoretical computer science and formal language theory, a regular tree grammar is a formal grammar that describes a set of directed trees, or terms. A regular word grammar can be seen as a special kind of regular tree grammar, describing a set of single-path trees. | Regular tree grammar | 0.782089 |
87,407 | Some of these particles will be guided by the Earth's field into the region where TRACERS can observe them. TRACERS will study a longstanding question about where reconnection happens at the magnetopause and how the solar wind affects the place and timing, helping NASA better forecast the influx of energetic particles ... | Tandem Reconnection and Cusp Electrodynamics Reconnaissance Satellites | 0.782086 |
87,408 | In the case of the group En(k) of elementary matrices, the universal central extension is now written Stn(k) and called the Steinberg group. In the spring of 1967, John Milnor defined K2(R) to be the kernel of the homomorphism St(R) → E(R). The group K2 further extended some of the exact sequences known for K1 and K0, ... | Algebraic K-theory | 0.782086 |
87,409 | In addition to providing a coherent exposition of the results then known, Bass improved many of the statements of the theorems. Of particular note is that Bass, building on his earlier work with Murthy, provided the first proof of what is now known as the fundamental theorem of algebraic K-theory. This is a four-term e... | Algebraic K-theory | 0.782086 |
87,410 | Motivated by this, the Bass–Schanuel definition of K1 of a ring R is GL(R) / E(R), where GL(R) is the infinite general linear group (the union of all GLn(R)) and E(R) is the subgroup of elementary matrices. They also provided a definition of K0 of a homomorphism of rings and proved that K0 and K1 could be fit together ... | Algebraic K-theory | 0.782086 |
87,411 | One possible definition of higher algebraic K-theory of rings was given by Quillen K n ( R ) = π n ( B GL ( R ) + ) , {\displaystyle K_{n}(R)=\pi _{n}(B\operatorname {GL} (R)^{+}),} Here πn is a homotopy group, GL(R) is the direct limit of the general linear groups over R for the size of the matrix tending to infinit... | Algebraic K-theory | 0.782086 |
87,412 | This space contains strictly more information than the Whitehead group; for example, the connected component of the trivial cobordism describes the possible cylinders on M and in particular is the obstruction to the uniqueness of a homotopy between a manifold and M × . Consideration of these questions led Waldhausen to... | Algebraic K-theory | 0.782086 |
87,413 | Jean Cerf proved that for simply connected smooth manifolds M of dimension at least 5, isotopy of h-cobordisms is the same as a weaker notion called pseudo-isotopy. Hatcher and Wagoner studied the components of the space of pseudo-isotopies and related it to a quotient of K2(Zπ).The proper context for the s-cobordism t... | Algebraic K-theory | 0.782086 |
87,414 | In fact the s-cobordism theorem implies that there is a bijective correspondence between isomorphism classes of h-cobordisms and elements of the Whitehead group. An obvious question associated with the existence of h-cobordisms is their uniqueness. The natural notion of equivalence is isotopy. | Algebraic K-theory | 0.782086 |
87,415 | If M and N are not assumed to be simply connected, then an h-cobordism need not be a cylinder. The s-cobordism theorem, due independently to Mazur, Stallings, and Barden, explains the general situation: An h-cobordism is a cylinder if and only if the Whitehead torsion of the inclusion M ⊂ W vanishes. This generalizes t... | Algebraic K-theory | 0.782086 |
87,416 | Two n-dimensional manifolds M and N are h-cobordant if there exists an (n + 1)-dimensional manifold with boundary W whose boundary is the disjoint union of M and N and for which the inclusions of M and N into W are homotopy equivalences (in the categories TOP, PL, or DIFF). Stephen Smale's h-cobordism theorem asserted ... | Algebraic K-theory | 0.782086 |
87,417 | The earliest application of algebraic K-theory to topology was Whitehead's construction of Whitehead torsion. A closely related construction was found by C. T. C. Wall in 1963. Wall found that a space X dominated by a finite complex has a generalized Euler characteristic taking values in a quotient of K0(Zπ), where π i... | Algebraic K-theory | 0.782086 |
87,418 | Hyman Bass provided this definition, which generalizes the group of units of a ring: K1(A) is the abelianization of the infinite general linear group: K 1 ( A ) = GL ( A ) ab = GL ( A ) / {\displaystyle K_{1}(A)=\operatorname {GL} (A)^{\mbox{ab}}=\operatorname {GL} (A)/} Here GL ( A ) = colim GL ( n , A ) {\... | Algebraic K-theory | 0.782086 |
87,419 | Matsumoto's theorem states that for a field k, the second K-group is given by K 2 ( k ) = k × ⊗ Z k × / ⟨ a ⊗ ( 1 − a ) ∣ a ≠ 0 , 1 ⟩ . {\displaystyle K_{2}(k)=k^{\times }\otimes _{\mathbf {Z} }k^{\times }/\langle a\otimes (1-a)\mid a\not =0,1\rangle .} Matsumoto's original theorem is even more general: For any root sy... | Algebraic K-theory | 0.782086 |
87,420 | In the mathematical area of graph theory, a chordal bipartite graph is a bipartite graph B = (X,Y,E) in which every cycle of length at least 6 in B has a chord, i.e., an edge that connects two vertices that are a distance > 1 apart from each other in the cycle. A better name would be weakly chordal and bipartite since ... | Chordal bipartite graph | 0.782085 |
87,421 | Her certificate of election reads: Jane Clarke is distinguished for the rigorous physical chemistry approaches she has adapted and applied to understand protein folding and misfolding. Her fundamental studies revealed the presence of both parallel pathways and frustration in the energy landscape of apparently simple pr... | Jane Clarke (scientist) | 0.782085 |
87,422 | She pioneered the application of protein engineering techniques together with single molecule force spectroscopy and simulations to investigate the effect of force on proteins. For this she was awarded the Biophysical Society U.S Genomics Award for Outstanding Investigator in the field of Single Molecule Biology in 201... | Jane Clarke (scientist) | 0.782085 |
87,423 | In 2010, Clarke was awarded the US Genomics award from the Biophysical Society. Clarke was also elected a Fellow of the Royal Society of Chemistry (FRSC) and a Fellow of the Academy of Medical Sciences (FMedSci) in 2013. Her nomination for the Academy of Medical Sciences in 2013 reads: Jane Clarke is a distinguished bi... | Jane Clarke (scientist) | 0.782085 |
87,424 | He obtained a job in the United States and the family moved there. Since Clarke was unable to work as a teacher, through not having appropriate qualifications she decided to update her scientific knowledge through a Master of Science degree in applied biology, awarded in 1990 from the Georgia Institute of Technology. T... | Jane Clarke (scientist) | 0.782085 |
87,425 | Clarke was born Jane Morgan in London on 10 September 1950. She was educated at the University of York where she graduated with a first-class honours degree in biochemistry in 1972. She went on to study for a Postgraduate Certificate in Education (PGCE) at the University of Cambridge in 1973. Clarke was a science teach... | Jane Clarke (scientist) | 0.782085 |
87,426 | Nature 474 662–665 Alessandro Borgia, Philip M. Williams and Jane Clarke (2008) Single-molecule studies of protein folding. Annual Review of Biochemistry 77 101–125 Jung-Hoon Han, Sarah Batey, Adrian A. Nickson, Sarah A. Teichmann and Jane Clarke (2007) The folding and evolution of multidomain proteins. Nature Reviews ... | Jane Clarke (scientist) | 0.782085 |
87,427 | Clarke was appointed a Wellcome Trust Senior Research Fellow in 2001, a Professor of Molecular Biophysics in 2009 and a Fellow of Trinity Hall, Cambridge, in 2010. On 1 October 2017, she became the President of Wolfson College, Cambridge.Clarke's research investigates protein folding, in particular: Studies of structur... | Jane Clarke (scientist) | 0.782085 |
87,428 | Jane Clarke (née Morgan; born 1950) is an English biochemist and academic. Since October 2017, she has served as President of Wolfson College, Cambridge. She is also Professor of Molecular Biophysics, a Wellcome Trust Senior Research Fellow in the Department of Chemistry at the University of Cambridge. She was previous... | Jane Clarke (scientist) | 0.782085 |
87,429 | Bleeping Computer is a website covering technology news and offering free computer help via its forums that was created by Lawrence Abrams in 2004. It publishes news focusing heavily on cybersecurity, but also covers other topics including computer software, computer hardware, operating system and general technology. I... | Bleeping Computer | 0.782085 |
87,430 | The articles published at Bleeping Computer are categorized as news articles, tutorials and virus removal guides. Its content includes searchable databases for looking up Windows start-up programs and uninstall entries, as well as a free Internet forum to receive computer help. The site covers news released by research... | Bleeping Computer | 0.782085 |
87,431 | The Small ExoLife Finder (SELF) telescope is a 3.5m outer diameter Fizeau telescope built from 15 0.5m diameter subapertures arranged in a circular pattern like the ELF telescope. SELF uses a telescope support structure built from a tensegrity structure of cables and compressional elements. This structure creates a sti... | ExoLife Finder | 0.782084 |
87,432 | The key to finding life is to measure reflected starlight from an exoplanet. ELF does this with a very large diameter telescope using optical technologies that can measure and correct the distortions in the optical wavefront created by the atmosphere above the telescope and the telescope's imperfections. The ELF depend... | ExoLife Finder | 0.782084 |
87,433 | The chemistry of the 1,2,3-triazol-5-ylidene system is a much more recently developed field. This system is based on the 1,2,3-triazole ring and had been indicated to have "non negligible lifetimes" in solution as early as 1975. In 2008, 1,2,3-triazolium iodide salts were observed to react with transition metals to for... | Triazol-5-ylidene | 0.782084 |
87,434 | Belyi's theorem is an existence theorem for Belyi functions, and has subsequently been much used in the inverse Galois problem. | Belyi's theorem | 0.782083 |
87,435 | A Belyi function is a holomorphic map from a compact Riemann surface S to the complex projective line P1(C) ramified only over three points, which after a Möbius transformation may be taken to be { 0 , 1 , ∞ } {\displaystyle \{0,1,\infty \}} . Belyi functions may be described combinatorially by dessins d'enfants. Belyi... | Belyi's theorem | 0.782083 |
87,436 | In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only. This is a result of G. V. Belyi from 1979. At the time it ... | Belyi's theorem | 0.782083 |
87,437 | The ring L { τ } {\displaystyle L\{\tau \}} can also be thought of as the ring of all (absolutely) additive polynomials a 0 x + a 1 x p + a 2 x p 2 + ⋯ = a 0 τ 0 + a 1 τ + a 2 τ 2 + ⋯ {\displaystyle a_{0}x+a_{1}x^{p}+a_{2}x^{p^{2}}+\cdots =a_{0}\tau ^{0}+a_{1}\tau +a_{2}\tau ^{2}+\cdots \,} in L {\displaystyle L} , wh... | Drinfeld module | 0.782082 |
87,438 | Laumon, Gérard (1996), Cohomology of Drinfeld Modular Varieties, Part 1, Geometry, Counting of Points and Local Harmonic Analysis, Cambridge Studies in Advanced Mathematics, vol. 41, Cambridge University Press, ISBN 978-0-521-47060-5 Laumon, Gérard; Waldspurger, Jean-Loup (1996), Cohomology of Drinfeld Modular Varietie... | Drinfeld module | 0.782082 |
87,439 | In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca) is a s... | Drinfeld module | 0.782082 |
87,440 | Many scientists were also reminded of the Mössbauer effect, a process involving nuclear transitions in a solid. Its discovery 30 years earlier had also been unexpected, though it was quickly replicated and explained within the existing physics framework.The announcement of a new purported clean source of energy came at... | Low energy nuclear reaction | 0.782082 |
87,441 | In mid-March 1989, both research teams were ready to publish their findings, and Fleischmann and Jones had agreed to meet at an airport on 24 March to send their papers to Nature via FedEx. Fleischmann and Pons, however, pressured by the University of Utah, which wanted to establish priority on the discovery, broke the... | Low energy nuclear reaction | 0.782082 |
87,442 | Let F be an algebraic function field with a finite field of constants and fix a place ∞ {\displaystyle \infty } of F. Define A to be the ring of elements in F that are regular at every place except possibly ∞ {\displaystyle \infty } . In particular, A is a Dedekind domain and it is discrete in F (with the topology indu... | Drinfeld module | 0.782082 |
87,443 | TPE materials that come from the block copolymers group include CAWITON†, MELIFLEX, THERMOLAST K†, THERMOLAST M†, Chemiton, Arnitel, Hytrel, Dryflex†, Mediprene, Kraton, Pibiflex, Sofprene†, Tuftec†and Laprene†.† indicates styrenic block copolymers (TPE-s). Laripur, Desmopan and Elastollan are examples of thermoplastic... | Thermoplastic elastomers | 0.782081 |
87,444 | Hence, batch-to-batch variations in weighting and metering components are absent, leading to improved consistency in both raw materials and fabricated articles. TPE materials have the potential to be recyclable since they can be molded, extruded and reused like plastics, but they have typical elastic properties of rubb... | Thermoplastic elastomers | 0.782081 |
87,445 | It is strongly emphasised that the system functions as an open platform and as the basis on which the actual ITS applications and services are built. The ITS Platform will develop four demonstration applications, which will be fully operational and may serve as inspiration for an actual product development by other int... | ITS Platform | 0.78208 |
87,446 | In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions. Equicontinuity appears in the fo... | Equicontinuous linear maps | 0.782078 |
87,447 | For example, if the sequence consists of differentiable functions or functions with some regularity (e.g., the functions are solutions of a differential equation), then the mean value theorem or some other kinds of estimates can be used to show the sequence is equicontinuous. It then follows that the limit of the seque... | Equicontinuous linear maps | 0.782077 |
87,448 | The hypothesis of the statement can be weakened a bit: a sequence in C(X) converges uniformly if it is equicontinuous and converges pointwise on a dense subset to some function on X (not assumed continuous). This weaker version is typically used to prove Arzelà–Ascoli theorem for separable compact spaces. Another conse... | Equicontinuous linear maps | 0.782077 |
87,449 | Let X be a compact Hausdorff space, and equip C(X) with the uniform norm, thus making C(X) a Banach space, hence a metric space. Then Arzelà–Ascoli theorem states that a subset of C(X) is compact if and only if it is closed, uniformly bounded and equicontinuous. This is analogous to the Heine–Borel theorem, which state... | Equicontinuous linear maps | 0.782077 |
87,450 | Like many categories, the category TVect is a concrete category, meaning its objects are sets with additional structure (i.e. a vector space structure and a topology) and its morphisms are functions preserving this structure. There are obvious forgetful functors into the category of topological spaces, the category of ... | Category of topological vector spaces | 0.782077 |
87,451 | But there is a catch: While the initial structure of the above property is in fact the usual initial topology on V {\displaystyle V} with respect to ( τ i , f i ) i ∈ I {\displaystyle (\tau _{i},f_{i})_{i\in I}} , the final structures do not need to be final with respect to given maps in the sense of Top. For example: ... | Category of topological vector spaces | 0.782077 |
87,452 | K {\displaystyle K} is a topological category The category is topological, which means loosely speaking that it relates to its "underlying category", the category of vector spaces, in the same way that Top relates to Set. Formally, for every K-vector space V {\displaystyle V} and every family ( ( V i , τ i ) , f i ) i ... | Category of topological vector spaces | 0.782077 |
87,453 | Negotiation between agents (software agents as well as humans) allows cooperative and competitive sharing of information to determine a proper price. Recent research and practice has also shown that electronic negotiation is beneficial for the coordination of complex interactions among organizations. Electronic negotia... | Social computing | 0.782076 |
87,454 | Collective intelligence is considered an area of social computing because of the group collaboration aspect. Becoming a growing area in computer science, collective intelligence provides users with a way to gain knowledge through collective efforts in a social interactive environment. | Social computing | 0.782076 |
87,455 | Social computing is an area of computer science that is concerned with the intersection of social behavior and computational systems. It is based on creating or recreating social conventions and social contexts through the use of software and technology. Thus, blogs, email, instant messaging, social network services, w... | Social computing | 0.782076 |
87,456 | Also in 1984, David A. Hounshell published From the American System to Mass Production, one of the most detailed histories of the machine tool industry from the late 18th century through 1932. It does not concentrate on listing firm names and sales statistics (which Floud's 1976 monograph focuses on) but rather is extr... | Machine tool builder | 0.782074 |
87,457 | To conclude, the multi-center Eguchi–Hanson geometry is a Kähler Ricci flat geometry which is asymptotically C2/Zn. According to Yau's theorem this is the only geometry satisfying these properties. Therefore, this is also the geometry of a C2/Zn orbifold in string theory after its conical singularity has been smoothed ... | Asymptotically locally Euclidean | 0.782072 |
87,458 | Unipotent groups over an algebraically closed field of any given dimension can in principle be classified, but in practice the complexity of the classification increases very rapidly with the dimension, so people tend to give up somewhere around dimension 6. | Unipotent | 0.782072 |
87,459 | This B-mode signal is generated at small angular scales by the gravitational lensing of the much larger primordial "E-mode" polarization signal (generated by scalar density perturbations at the time the CMB was emitted) and at large angular scales by the interaction of the CMB with a background of gravitational waves p... | South Pole Telescope | 0.782072 |
87,460 | The South Pole Telescope is funded through the National Science Foundation Office of Polar Programs and the United States Department of Energy, with additional support from the Kavli Foundation and the Gordon and Betty Moore Foundation. Funding for the SPTpol and SPT-3G instruments and operations are also provided by t... | South Pole Telescope | 0.782072 |
87,461 | When the characteristic of the base field is p there is an analogous statement for an algebraic group G {\displaystyle G}: there exists a smallest subgroup H {\displaystyle H} such that G / H {\displaystyle G/H} is a unipotent group H {\displaystyle H} is an extension of an abelian variety A {\displaystyle A} by a grou... | Unipotent | 0.782072 |
87,462 | The unipotent radical of an algebraic group G is the set of unipotent elements in the radical of G. It is a connected unipotent normal subgroup of G, and contains all other such subgroups. A group is called reductive if its unipotent radical is trivial. If G is reductive then its radical is a torus. | Unipotent | 0.782072 |
87,463 | Any unipotent algebraic group is isomorphic to a closed subgroup of the group of upper triangular matrices with diagonal entries 1, and conversely any such subgroup is unipotent. In particular any unipotent group is a nilpotent group, though the converse is not true (counterexample: the diagonal matrices of GLn(k)). Fo... | Unipotent | 0.782072 |
87,464 | An element x of an affine algebraic group is unipotent when its associated right translation operator, rx, on the affine coordinate ring A of G is locally unipotent as an element of the ring of linear endomorphism of A. (Locally unipotent means that its restriction to any finite-dimensional stable subspace of A is unip... | Unipotent | 0.782072 |
87,465 | Over characteristic 0 there is a nice classification of unipotent algebraic groups with respect to nilpotent Lie algebras. Recall that a nilpotent Lie algebra is a subalgebra of some g l n {\displaystyle {\mathfrak {gl}}_{n}} such that the iterated adjoint action eventually terminates to the zero-map. In terms of matri... | Unipotent | 0.782072 |
87,466 | Urbanization combined with increasing availability of convenient, safe and efficient public transport, green buildings and efficient energy distribution, as well as extended product life/use/re-use, increased local recycling and self-sustainability in raw materials drive down energy consumption. Perversely, ready acces... | Carbon bubble | 0.78207 |
87,467 | As the penetration of the renewable energy increases so will the wealth of the renewable energy corporations. This and the increasing number of employees in the renewable energy sector will inevitably transform into political lobbying against fossil fuels. | Carbon bubble | 0.78207 |
87,468 | The price of renewable energy is continually dropping. As of 2014 new wind power is cheaper than new coal and gas power in Australia, China and the United States. Also the electricity produced from a photovoltaic roof system is cheaper than the electricity from the grid in many countries and places in the world. | Carbon bubble | 0.78207 |
87,469 | The carbon bubble is a hypothesized bubble in the valuation of companies dependent on fossil-fuel-based energy production, resulting from future decreases in value of fossil fuel reserves as they become unusable in order to meet carbon budgets and recognition of negative externalities of carbon fuels which are not yet ... | Carbon bubble | 0.78207 |
87,470 | Innovations in, for example, information technology, miniaturisation, LEDs, virtual reality, 3D printing, new materials and biotechnology enable energy reduction in the areas of human sustenance and travel, as well as physical product creation and distribution. They also offer new avenues for economic growth and techno... | Carbon bubble | 0.78207 |
87,471 | In population ecology, Moran's theorem (or the Moran effect) states that the time correlation of two separate populations of the same species is equal to the correlation between the environmental variabilities where they live. The theorem is named after Pat Moran, who stated it in a paper on the dynamics of the Canadia... | Moran's theorem | 0.782069 |
87,472 | Photometric redshifts were originally determined by calculating the expected observed data from a known emission spectrum at a range of redshifts. The technique relies upon the spectrum of radiation being emitted by the object having strong features that can be detected by the relatively crude filters. As photometric f... | Photometric redshift | 0.782068 |
87,473 | Arctic tundra, which lined the north of the mammoth steppe, reflected modern ecology with species such as the polar bear, wolf, reindeer and muskox. Other biomes, although less noted, were significant in contributing to the diversity of fauna in Late Pleistocene Europe. Warmer grasslands such as temperate steppe and Me... | Overkill hypothesis | 0.782067 |
87,474 | Problem-Oriented Software Engineering provides a formal definition of problems, and a framework for associating problems with solutions through formal, logical arguments; Problem Frames provide a framework for defining empirical models of software engineering problems which are grounded in the physical world. The Probl... | Problem-oriented development | 0.782067 |
87,475 | Problem-Oriented Development is an emerging paradigm of computing that emphasises problems (as opposed to requirements) as the primary subject of scrutiny by software engineers. As such, Problem-Oriented Development is concerned with: Investigating the structure of organisational problems as addressed by Software Engin... | Problem-oriented development | 0.782067 |
87,476 | Knowledge Engineering is a field of computer science which offers a significant body of work on problem-solving methods. Problem-solving methods are patterns of reasoning that are applied to solve specific problems. The work on problem abstractions within knowledge engineering can therefore provide a basis for research... | Problem-oriented development | 0.782067 |
87,477 | In 2013, researchers at the University of Colorado Boulder announced the development of a solid-state lithium battery, with a solid composite cathode based on an iron-sulfur chemistry, that promised higher energy capacity compared to already-existing SSBs.In 2017, John Goodenough, the co-inventor of Li-ion batteries, u... | Solid-state battery | 0.782064 |
87,478 | Its performance and efficiency are constrained by the nature of its geometry. The current drawn from a thin film battery largely depends on the geometry and interface contacts of the electrolyte/cathode and the electrolyte/anode interfaces Low thickness of the electrolyte and the interfacial resistance at the electrode... | Solid-state battery | 0.782064 |
87,479 | #3 – Intersection of two lines This construction can also be done directly with a straightedge. #4, #5 - The other constructions Thus, to prove the theorem, only constructions #4 and #5 need be proven possible using only a straightedge and a given circle with its center. Notes Any doubts about constructions #1 or #3 wo... | Poncelet–Steiner theorem | 0.782064 |
87,480 | A circle is considered to be given by any two points, one defining the center and one existing on the circumference at radius. Any such pair define a unique circle. In keeping with the intent of the theorem which we aim to prove, the actual circle need not be drawn but for aesthetic reasons. | Poncelet–Steiner theorem | 0.782064 |
87,481 | To prove the theorem, each of the basic constructions of compass and straightedge need to be proven to be possible by using a straightedge alone (provided that a circle and its center exist in the plane), as these are the foundations of, or elementary steps for, all other constructions. That is to say, all construction... | Poncelet–Steiner theorem | 0.782064 |
87,482 | The Poncelet-Steiner Theorem does not prohibit the normal treatment of circles already drawn in the plane; normal construction rules apply. The theorem only prohibits the construction of new circular arcs with a compass. Steiner constructions and those constructions herein proving the Poncelet-Steiner theorem require t... | Poncelet–Steiner theorem | 0.782064 |
87,483 | Per the theorem, when a compass-drawn circle is provided it is simply referred to as the given circle or the provided circle. The provided circle should always be assumed to be placed arbitrarily in the plane with an arbitrary radius (i.e. in general position). The intersection points between any line and the given cir... | Poncelet–Steiner theorem | 0.782064 |
87,484 | This groundwork was provided by Jean Victor Poncelet in 1822, having been motivated by Mohr's work on the Mohr-Mascheroni theorem. He also conjectured and suggested a possible proof that a straightedge and rusty compass would be equivalent to a straightedge and compass, and moreover, the rusty compass need only be used... | Poncelet–Steiner theorem | 0.782064 |
87,485 | All points that uniquely define a construction, which can be determined with the use of the compass, are equally determinable without. Constructions carried out in adherence with this theorem - relying solely on the use of a straightedge tool without the aid of a compass - are known as Steiner constructions. Steiner co... | Poncelet–Steiner theorem | 0.782064 |
87,486 | In the branch of mathematics known as Euclidean geometry, the Poncelet–Steiner theorem is one of several results concerning compass and straightedge constructions having additional restrictions imposed on the traditional rules. This result states that whatever can be constructed by straightedge and compass together can... | Poncelet–Steiner theorem | 0.782064 |
87,487 | Various other notions, tools, terminology, etc., is often associated (sometimes loosely) to the Poncelet-Steiner theorem. Some are listed here. | Poncelet–Steiner theorem | 0.782064 |
87,488 | This construction does require the use of the given circle. In order to generalize the parallel line construction to all possible lines, not just the ones with a collinear bisected line segment, it becomes necessary to have additional information. In keeping with the Poncelet-Steiner theorem, a circle (with center) is ... | Poncelet–Steiner theorem | 0.782064 |
87,489 | This construction does require the use of the given circle and takes advantage of Thales's theorem. From a given line m, and a given point A in the plane, a perpendicular to the line is to be constructed through the point. Provided is the given circle O(r). If the desired line from which a perpendicular is to be made, ... | Poncelet–Steiner theorem | 0.782064 |
87,490 | The term Steiner construction typically refers to any geometric construction that utilizes the straightedge tool only, and is sometimes simply called a straightedge-only construction. No stipulations are made about what geometric objects already exist in the plane, and no implications are made about what is or is not p... | Poncelet–Steiner theorem | 0.782064 |
87,491 | The second basic construction - defining a circle with two points - never needed an arc to be constructed with the compass in order for the circle to be utilized in constructions - namely the intersections with circles and with lines which, together, are the essence of all constructions involving a circle. Thus definin... | Poncelet–Steiner theorem | 0.782064 |
87,492 | The rusty compass, therefore, is no weaker than the Poncelet-Steiner theorem. The rusty compass is also no stronger. The Poncelet-Steiner theorem reduces Ferrari's rusty compass equivalence, a claim at the time, to a single-use compass: all points necessary to uniquely describe any compass-straightedge construction may... | Poncelet–Steiner theorem | 0.782064 |
87,493 | Naturally if all constructions are possible with a single circle arbitrarily placed in the plane, then the same can surely be said about a straightedge and rusty compass. It is known that a straightedge and a rusty compass is sufficient to construct all that is possible with straightedge and standard compass - with the... | Poncelet–Steiner theorem | 0.782064 |
87,494 | The Poncelet-Steiner theorem requires a single circle with arbitrary radius and center point to be placed in the plane. As it is the only drawn circle, whether or not it was drawn by a rusty compass is immaterial and equivalent. The benefit of general rusty compass constructions, however, is that the compass may be use... | Poncelet–Steiner theorem | 0.782064 |
87,495 | As a restricted construction paradigm, the rusty compass constructions allow the use of a straightedge and the fixed-width compass. In some sense, the rusty compass is a generalization and simplification of the Poncelet-Steiner theorem. Though not more powerful, it is certainly more convenient. | Poncelet–Steiner theorem | 0.782064 |
87,496 | The inclusion of this sufficient alternative information disambiguates the mappings under the projective transformations, thus allowing various Steiner constructions to recover the circle center. Some alternatives include two concentric or two intersecting circles, or three circles, or other variations wherein the prov... | Poncelet–Steiner theorem | 0.782064 |
87,497 | Alternatively, the center may be omitted with sufficient additional information. This is not a weakening of the Poncelet-Steiner theorem, merely an alternative framework. Nor is it a contradiction of Steiner's Theorem which hypothesizes only a single circle. | Poncelet–Steiner theorem | 0.782064 |
87,498 | Thus it is not possible to construct everything that can be constructed with straightedge and compass with straightedge alone. Consequently, requirements on the Poncelet-Steiner theorem cannot be weakened with respect to the circle center. If the centre of the only given circle is not provided, it cannot be obtained by... | Poncelet–Steiner theorem | 0.782064 |
87,499 | If only one circle is to be given and no other special information, Steiner's theorem implies that the center of the circle must be provided along with the circle. This is done by proving the impossibility of constructing the circle's center from straightedge alone using only a single circle in the plane, without its c... | Poncelet–Steiner theorem | 0.782064 |
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