id
int32 0
100k
| text
stringlengths 21
3.54k
| source
stringlengths 1
124
| similarity
float32 0.78
0.88
|
|---|---|---|---|
99,900
|
It provided full pipeline delay management and optimization, and a consistent API for programming imaging hardware. Key software programmers were brought on: Ken Woodland, Stephen Watkins and Ari Berman. Recognizing that not every imaging function could be best done in a pipeline, Siegel teamed with Analog Devices new digital signal processor (DSP) group to develop Euclid, based on the ADSP-2100.
|
Datacube Inc.
| 0.776525
|
99,901
|
A Modular and expandable system based on the VMEbus form factor could meet many customer needs. MaxVideo and the MaxBus were born. Marketing research determined the primary functions required and a road map for the next few years. The first seven MaxVideo boards were Digimax (digitizer and display), Framestore (triple 512^2 framestore with unprecedented density), VFIR (first real-time 3x3 image filter, SNAP (3x3 Systolic Neighborhood Array Processor), Featuremax (real-time statistics) SP (single point general purpose processor) and Protomax (MaxVideo prototyping board).
|
Datacube Inc.
| 0.776525
|
99,902
|
For its first two years, CSET plans to focus on the intersection of security and artificial intelligence (AI), particularly on national competitiveness, talent and knowledge flows and relationships with other technologies. CSET is the largest center in the U.S. focused on AI and policy.
|
Center for Security and Emerging Technology
| 0.776523
|
99,903
|
On the other hand, by the special theorem for compact operators, H is the orthogonal direct sum of the eigenspaces of A, all finite-dimensional except possibly for the 0 eigenspace. Since RA = K* R, the image under R of the λ eigenspace of A lies in the λ eigenspace of K*. Similarly R carries the λ eigenspace of K into the λ eigenspace of A. It follows that the eigenvalues of K and K* are all real.
|
Symmetrizable compact operator
| 0.776522
|
99,904
|
The generalized version of this theorem was first proved by Kazlow (1979), also proved by Boivin and Dwilewicz (1998) under more less complicated hypothese. Theorem Let ω {\displaystyle \omega } be a connected submanifold of R n {\displaystyle \mathbb {R} ^{n}} of class- C 2 {\displaystyle C^{2}} . Then every continuous CR function on the tube domain Ω ( ω ) {\displaystyle \Omega (\omega )} can be continuously extended to a CR function on Ω ( ach ( ω ) ) .
|
Bochner's tube theorem
| 0.77652
|
99,905
|
In mathematics, Bochner's tube theorem (named for Salomon Bochner) shows that every function holomorphic on a tube domain in C n {\displaystyle \mathbb {C} ^{n}} can be extended to the convex hull of this domain. Theorem Let ω ⊂ R n {\displaystyle \omega \subset \mathbb {R} ^{n}} be a connected open set. Then every function f ( z ) {\displaystyle f(z)} holomorphic on the tube domain Ω = ω + i R n {\displaystyle \Omega =\omega +i\mathbb {R} ^{n}} can be extended to a function holomorphic on the convex hull ch ( Ω ) {\displaystyle \operatorname {ch} (\Omega )} . A classic reference is (Theorem 9). See also for other proofs.
|
Bochner's tube theorem
| 0.77652
|
99,906
|
The Triple Phase Boundary (TPB) length, which is the length where porous, ionic and electronically conducting pathways all meet, directly relates to the electrochemically active length in the cell. The larger the length, the more reactions can occur and thus the less the activation polarization. Optimization of TPB length can be done by processing conditions to affect microstructure or by materials selection to use a mixed ionic/electronic conductor to further increase TPB length.
|
Solid Oxide Fuel Cell
| 0.776519
|
99,907
|
The activation polarization is the result of the kinetics involved with the electrochemical reactions. Each reaction has a certain activation barrier that must be overcome in order to proceed and this barrier leads to the polarization. The activation barrier is the result of many complex electrochemical reaction steps where typically the rate limiting step is responsible for the polarization. The polarization equation shown below is found by solving the Butler–Volmer equation in the high current density regime (where the cell typically operates), and can be used to estimate the activation polarization: η a c t = R T β z F × l n ( i i 0 ) {\displaystyle {\eta }_{act}={\frac {RT}{{\beta }zF}}\times ln\left({\frac {i}{{i}_{0}}}\right)} where: R {\displaystyle R} = gas constant T 0 {\displaystyle {T}_{0}} = operating temperature β {\displaystyle {\beta }} = electron transfer coefficient z {\displaystyle z} = electrons associated with the electrochemical reaction F {\displaystyle F} = Faraday's constant i {\displaystyle i} = operating current i 0 {\displaystyle i_{0}} = exchange current densityThe polarization can be modified by microstructural optimization.
|
Solid Oxide Fuel Cell
| 0.776519
|
99,908
|
Unlike most other types of fuel cells, SOFCs can have multiple geometries. The planar fuel cell design geometry is the typical sandwich type geometry employed by most types of fuel cells, where the electrolyte is sandwiched in between the electrodes. SOFCs can also be made in tubular geometries where either air or fuel is passed through the inside of the tube and the other gas is passed along the outside of the tube.
|
Solid Oxide Fuel Cell
| 0.776519
|
99,909
|
Thermal expansion demands a uniform and well-regulated heating process at startup. SOFC stacks with planar geometry require on the order of an hour to be heated to operating temperature. Micro-tubular fuel cell design geometries promise much faster start up times, typically in the order of minutes.
|
Solid Oxide Fuel Cell
| 0.776519
|
99,910
|
Also, the 3D printing technique allows the cell layers to be printed on top of each other instead of having to go through separate manufacturing and stacking steps. The thickness is easy to control, and layers can be made in the exact size and shape that is needed, so waste is minimized.Ceres Power Ltd. has developed a low cost and low temperature (500–600 degrees) SOFC stack using cerium gadolinium oxide (CGO) in place of current industry standard ceramic, yttria stabilized zirconia (YSZ), which allows the use of stainless steel to support the ceramic.Solid Cell Inc. has developed a unique, low-cost cell architecture that combines properties of planar and tubular designs, along with a Cr-free cermet interconnect. The high temperature electrochemistry center (HITEC) at the University of Florida, Gainesville is focused on studying ionic transport, electrocatalytic phenomena and microstructural characterization of ion conducting materials.SiEnergy Systems, a Harvard spin-off company, has demonstrated the first macro-scale thin-film solid-oxide fuel cell that can operate at 500 degrees.
|
Solid Oxide Fuel Cell
| 0.776519
|
99,911
|
This manufacturing technique would allow SOFC cell structure to be more flexible, which could lead to more efficient designs. This process could work in the production of any part of the cell. The 3D printing process works by combining about 80% ceramic particles with 20% binders and solvents, and then converting that slurry into an ink that can be fed into a 3D printer.
|
Solid Oxide Fuel Cell
| 0.776519
|
99,912
|
A high-temperature SOFC will generate all of the needed electricity to allow the engine to be smaller and more efficient. The SOFC would run on the same gasoline or diesel as the engine and would keep the air conditioning unit and other necessary electrical systems running while the engine shuts off when not needed (e.g., at a stop light or truck stop).Rolls-Royce is developing solid-oxide fuel cells produced by screen printing onto inexpensive ceramic materials. Rolls-Royce Fuel Cell Systems Ltd is developing an SOFC gas turbine hybrid system fueled by natural gas for power generation applications in the order of a megawatt (e.g. Futuregen).3D printing is being explored as a possible manufacturing technique that could be used to make SOFC manufacturing easier by the Shah Lab at Northwestern University.
|
Solid Oxide Fuel Cell
| 0.776519
|
99,913
|
The final target of study for this group was the medical students they taught. Medical students at Ohio State University, Ohio University and Northeast Ohio Medical College were invited to serve as subjects. Students indicating a preference for the specialties of family practice, psychiatry, pediatrics and obstetrics-gynecology achieved significantly higher levels of accuracy than those students who planned to train as surgeons, radiologists, or pathologists. Internal medicine and plastic surgery candidates scored at levels near the mean.
|
Non verbal communication
| 0.776519
|
99,914
|
In mathematics, the Andreotti–Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel (1959), states that if V {\displaystyle V} is a smooth, complex affine variety of complex dimension n {\displaystyle n} or, more generally, if V {\displaystyle V} is any Stein manifold of dimension n {\displaystyle n} , then V {\displaystyle V} admits a Morse function with critical points of index at most n, and so V {\displaystyle V} is homotopy equivalent to a CW complex of real dimension at most n. Consequently, if V ⊆ C r {\displaystyle V\subseteq \mathbb {C} ^{r}} is a closed connected complex submanifold of complex dimension n {\displaystyle n} , then V {\displaystyle V} has the homotopy type of a CW complex of real dimension ≤ n {\displaystyle \leq n} . Therefore H i ( V ; Z ) = 0 , for i > n {\displaystyle H^{i}(V;\mathbb {Z} )=0,{\text{ for }}i>n} and H i ( V ; Z ) = 0 , for i > n . {\displaystyle H_{i}(V;\mathbb {Z} )=0,{\text{ for }}i>n.} This theorem applies in particular to any smooth, complex affine variety of dimension n {\displaystyle n} .
|
Andreotti–Frankel theorem
| 0.776519
|
99,915
|
The society meets regularly on Thursday evenings in the Godlee Observatory, with the exception of Public Lecture evenings (see below) and the Christmas closure of the University building. These meetings consist of observing using the telescopes (weather permitting), informal talks and discussion sessions.The society hosts public lectures on astronomy on the third Thursday of the month between October and March in the John Dalton Building, Manchester Metropolitan University, Oxford Road, Manchester. They are generally given by invited speakers although the December lecture is traditionally given by the President of the society and is held in the Godlee Observatory. The society holds its Annual General Meeting on the third Thursday of April in the Godlee Observatory.
|
Manchester Astronomical Society
| 0.776519
|
99,916
|
Using 3D printing and multi-material structures in additive manufacturing has allowed for the design and creation of what is called 4D printing. 4D printing is an additive manufacturing process in which the printed object changes shape with time, temperature, or some other type of stimulation. 4D printing allows for the creation of dynamic structures with adjustable shapes, properties or functionality.
|
3D Printer
| 0.776519
|
99,917
|
It is therefore an additional object of the invention to minimize use to materials in a process of the indicated class. It is a further object of the invention that materials employed in such a process be salvaged for reuse. According to another aspect of the invention, a combination for writing and the like comprises a carrier for displaying an intelligence pattern and an arrangement for removing the pattern from the carrier. In 1974, David E. H. Jones laid out the concept of 3D printing in his regular column Ariadne in the journal New Scientist.
|
3D Printer
| 0.776519
|
99,918
|
In 1971, Johannes F Gottwald patented the Liquid Metal Recorder, U.S. Patent 3596285A, a continuous inkjet metal material device to form a removable metal fabrication on a reusable surface for immediate use or salvaged for printing again by remelting. This appears to be the first patent describing 3D printing with rapid prototyping and controlled on-demand manufacturing of patterns. The patent states: As used herein the term printing is not intended in a limited sense but includes writing or other symbols, character or pattern formation with an ink.
|
3D Printer
| 0.776519
|
99,919
|
Computed axial lithography is a method for 3D printing based on computerised tomography scans to create prints in photo-curable resin. It was developed by a collaboration between the University of California, Berkeley with Lawrence Livermore National Laboratory. Unlike other methods of 3D printing it does not build models through depositing layers of material like fused deposition modelling and stereolithography, instead it creates objects using a series of 2D images projected onto a cylinder of resin. It is notable for its ability to build an object much more quickly than other methods using resins and the ability to embed objects within the prints.
|
3D Printer
| 0.776519
|
99,920
|
The entrance of the building was chosen due to the 3D printing limitations and the budget of the project for producing the maquette. 3D printing was only one of the capabilities enabled by the produced 3D model of the bank, but due to the project's limited scope, the team did not continue modelling for the virtual representation or other applications. In 2021, Parsinejad et al. comprehensively compared the hand surveying method for 3D reconstruction ready for 3D printing with digital recording (adoption of photogrammetry method).
|
3D Printer
| 0.776519
|
99,921
|
The application of 3D printing for the representation of architectural assets has many challenges. In 2018, the structure of Iran National Bank was traditionally surveyed and modeled in computer graphics software (specifically, Cinema4D) and was optimized for 3D printing. The team tested the technique for the construction of the part and it was successful. After testing the procedure, the modellers reconstructed the structure in Cinema4D and exported the front part of the model to Netfabb.
|
3D Printer
| 0.776519
|
99,922
|
The Manchester Astronomical Society is an organisation that promotes popular and amateur astronomy in North West England. It is one of the oldest provincial astronomical societies in England. The Society is based in the Godlee Observatory located in the Sackville Building, University of Manchester, in Manchester city centre. Membership is open to anyone with an interest in astronomy.
|
Manchester Astronomical Society
| 0.776519
|
99,923
|
Inventory data is imported and maintained from the schedule distribution system. Changes to aircraft availability would immediately impact the available seats of the fleet, as well as the seats which had been sold. The price for each sold seat is determined by a combination of the fares and booking conditions stored in the Fare Quote System,. In most cases, inventory control has a real time interface to an airline's yield management system to support a permanent optimization of the offered booking classes in response to changes in demand or pricing strategies of competitors.
|
Flight reservation system
| 0.776518
|
99,924
|
By the change of variables t = −ln s, this becomes the familiar Euler integral. Euler published his results in the paper "De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt" ("On transcendental progressions, that is, those whose general terms cannot be given algebraically"), submitted to the St.
|
Euler Gamma Function
| 0.776518
|
99,925
|
A more restrictive property than satisfying the above interpolation is to satisfy the recurrence relation defining a translated version of the factorial function, for any positive real number x. But this would allow for multiplication by any function g(x) satisfying both g(x) = g(x+1) for all real numbers x and g(0) = 1, such as the function g(x) = e k sin 2mπx. One of several ways to resolve the ambiguity comes from the Bohr–Mollerup theorem. It states that when the condition that f be logarithmically convex (or "super-convex", meaning that ln ∘ f {\displaystyle \ln \circ f} is convex) is added, it uniquely determines f for positive, real inputs. From there, the gamma function can be extended to all real and complex values (except the negative integers and zero) by using the unique analytic continuation of f.
|
Euler Gamma Function
| 0.776518
|
99,926
|
There are, relatively speaking, no such simple solutions for factorials; no finite combination of sums, products, powers, exponential functions, or logarithms will suffice to express x! ; but it is possible to find a general formula for factorials using tools such as integrals and limits from calculus. A good solution to this is the gamma function.There are infinitely many continuous extensions of the factorial to non-integers: infinitely many curves can be drawn through any set of isolated points.
|
Euler Gamma Function
| 0.776518
|
99,927
|
Using the pi function the reflection formula takes on the form where sinc is the normalized sinc function, while the multiplication theorem takes on the form We also sometimes find which is an entire function, defined for every complex number, just like the reciprocal gamma function. That π ( z ) {\displaystyle \pi (z)} is entire entails it has no poles, so Π ( z ) {\displaystyle \Pi \left(z\right)} , like Γ ( z ) {\displaystyle \Gamma \left(z\right)} , has no zeros. The volume of an n-ellipsoid with radii r1, …, rn can be expressed as
|
Euler Gamma Function
| 0.776518
|
99,928
|
Double-precision floating-point implementations of the gamma function and its logarithm are now available in most scientific computing software and special functions libraries, for example TK Solver, Matlab, GNU Octave, and the GNU Scientific Library. The gamma function was also added to the C standard library (math.h). Arbitrary-precision implementations are available in most computer algebra systems, such as Mathematica and Maple. PARI/GP, MPFR and MPFUN contain free arbitrary-precision implementations. In some software calculators, e.g. Windows Calculator and GNOME Calculator, the factorial function returns Γ(x+1) when the input x is a non-integer value.
|
Euler Gamma Function
| 0.776518
|
99,929
|
"There was in fact little practical need for anything but real values of the gamma function until the 1930s, when applications for the complex gamma function were discovered in theoretical physics. As electronic computers became available for the production of tables in the 1950s, several extensive tables for the complex gamma function were published to meet the demand, including a table accurate to 12 decimal places from the U.S. National Bureau of Standards.
|
Euler Gamma Function
| 0.776518
|
99,930
|
The Bohr–Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the gamma function is log-convex, that is, its natural logarithm is convex on the positive real axis. Another characterisation is given by the Wielandt theorem. The gamma function is the unique function that simultaneously satisfies Γ ( 1 ) = 1 {\displaystyle \Gamma (1)=1} , Γ ( z + 1 ) = z Γ ( z ) {\displaystyle \Gamma (z+1)=z\Gamma (z)} for all complex numbers z {\displaystyle z} except the non-positive integers, and, for integer n, lim n → ∞ Γ ( n + z ) Γ ( n ) n z = 1 {\textstyle \lim _{n\to \infty }{\frac {\Gamma (n+z)}{\Gamma (n)\;n^{z}}}=1} for all complex numbers z {\displaystyle z} .In a certain sense, the ln(Γ) function is the more natural form; it makes some intrinsic attributes of the function clearer. A striking example is the Taylor series of ln(Γ) around 1: with ζ(k) denoting the Riemann zeta function at k. So, using the following property: we can find an integral representation for the ln(Γ) function: or, setting z = 1 to obtain an integral for γ, we can replace the γ term with its integral and incorporate that into the above formula, to get: There also exist special formulas for the logarithm of the gamma function for rational z. For instance, if k {\displaystyle k} and n {\displaystyle n} are integers with k < n {\displaystyle k
|
Euler Gamma Function
| 0.776518
|
99,931
|
However, these numbers Γ ( r ) {\displaystyle \Gamma (r)} are not known to be expressible by themselves in terms of elementary functions. It has been proved that Γ ( n + r ) {\displaystyle \Gamma (n+r)} is a transcendental number and algebraically independent of π {\displaystyle \pi } for any integer n {\displaystyle n} and each of the fractions r = 1 6 , 1 4 , 1 3 , 2 3 , 3 4 , 5 6 {\textstyle r={\frac {1}{6}},{\frac {1}{4}},{\frac {1}{3}},{\frac {2}{3}},{\frac {3}{4}},{\frac {5}{6}}} . In general, when computing values of the gamma function, we must settle for numerical approximations.
|
Euler Gamma Function
| 0.776518
|
99,932
|
!=(2n-1)(2n-3)\cdots (3)(1)} . See Particular values of the gamma function for calculated values. It might be tempting to generalize the result that Γ ( 1 2 ) = π {\textstyle \Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }}} by looking for a formula for other individual values Γ ( r ) {\displaystyle \Gamma (r)} where r {\displaystyle r} is rational, especially because according to Gauss's digamma theorem, it is possible to do so for the closely related digamma function at every rational value.
|
Euler Gamma Function
| 0.776518
|
99,933
|
Other important functional equations for the gamma function are Euler's reflection formula which implies and the Legendre duplication formula The duplication formula is a special case of the multiplication theorem (see Eq. 5.5.6): A simple but useful property, which can be seen from the limit definition, is: In particular, with z = a + bi, this product is If the real part is an integer or a half-integer, this can be finitely expressed in closed form: Perhaps the best-known value of the gamma function at a non-integer argument is which can be found by setting z = 1 2 {\textstyle z={\frac {1}{2}}} in the reflection or duplication formulas, by using the relation to the beta function given below with z 1 = z 2 = 1 2 {\textstyle z_{1}=z_{2}={\frac {1}{2}}} , or simply by making the substitution u = z {\displaystyle u={\sqrt {z}}} in the integral definition of the gamma function, resulting in a Gaussian integral. In general, for non-negative integer values of n {\displaystyle n} we have: where the double factorial ( 2 n − 1 ) ! ! = ( 2 n − 1 ) ( 2 n − 3 ) ⋯ ( 3 ) ( 1 ) {\displaystyle (2n-1)!
|
Euler Gamma Function
| 0.776518
|
99,934
|
The Bohr–Mollerup theorem is useful because it is relatively easy to prove logarithmic convexity for any of the different formulas used to define the gamma function. Taking things further, instead of defining the gamma function by any particular formula, we can choose the conditions of the Bohr–Mollerup theorem as the definition, and then pick any formula we like that satisfies the conditions as a starting point for studying the gamma function. This approach was used by the Bourbaki group. Borwein & Corless review three centuries of work on the gamma function.
|
Euler Gamma Function
| 0.776518
|
99,935
|
A definite and generally applicable characterization of the gamma function was not given until 1922. Harald Bohr and Johannes Mollerup then proved what is known as the Bohr–Mollerup theorem: that the gamma function is the unique solution to the factorial recurrence relation that is positive and logarithmically convex for positive z and whose value at 1 is 1 (a function is logarithmically convex if its logarithm is convex). Another characterisation is given by the Wielandt theorem.
|
Euler Gamma Function
| 0.776518
|
99,936
|
However, the gamma function does not appear to satisfy any simple differential equation. Otto Hölder proved in 1887 that the gamma function at least does not satisfy any algebraic differential equation by showing that a solution to such an equation could not satisfy the gamma function's recurrence formula, making it a transcendentally transcendental function. This result is known as Hölder's theorem.
|
Euler Gamma Function
| 0.776518
|
99,937
|
Instead of finding a specialized proof for each formula, it would be desirable to have a general method of identifying the gamma function. One way to prove would be to find a differential equation that characterizes the gamma function. Most special functions in applied mathematics arise as solutions to differential equations, whose solutions are unique.
|
Euler Gamma Function
| 0.776518
|
99,938
|
Infectious diseases spread across the globe usually because of long-distance travels of carriers of the disease. These long-distance travels are made using air transportation systems and it has been shown that "network topology, traffic structure, and individual mobility patterns are all essential for accurate predictions of disease spreading". On a smaller spatial scale the regularity of human movement patterns and its temporal structure should be taken into account in models of infectious disease spread. Cellphone viruses that are transmitted via bluetooth are greatly dependent on the human interaction and movements. With more people using similar operating systems for their cellphones, it's becoming much easier to have a virus epidemic.In Transportation Planning, leveraging the characteristics of human movement, such as tendency to travel short distances with few but regular bursts of long-distance trips, novel improvements have been made to Trip distribution models, specifically to Gravity model of migration
|
Individual mobility
| 0.776518
|
99,939
|
Some recent scientific studies compared the mobility patterns emerged from mobile phone data with those emerged from GPS data.Researchers have been able to extract very detailed information about the people whose data are made available to public. This has sparked a great amount of concern about privacy issues. As an example of liabilities that might happen, New York City released 173 million individual taxi trips. City officials used a very weak cryptography algorithm to anonymize the license number and medallion number, which is an alphanumeric code assigned to each taxi cab. This made it possible for hackers to completely de-anonymize the dataset, and even some were able to extract detailed information about specific passengers and celebrities, including their origin and destination and how much they tipped.
|
Individual mobility
| 0.776518
|
99,940
|
OGLE-2016-BLG-1195Lb is an extrasolar planet located about 22,000 light-years from Earth, in the galactic bulge, orbiting the 0.57±0.06M☉ star OGLE-2016-BLG-1195L, discovered in 2017. The planet was detected using gravitational microlensing techniques managed by the Korea Astronomy and Space Science Institute and the Spitzer Space Telescope. Initially, it was believed the planet has a mass similar to Earth and is located about the same distance from its host star as the Earth is from the Sun, although it was expected to be much colder.In 2023, the planetary mass estimate was significantly increased to 9.91±1.61M🜨, resulting in it likely being a Neptune-like ice giant planet.
|
OGLE-2016-BLG-1195Lb
| 0.776517
|
99,941
|
{\displaystyle L^{2}\left(X;L^{2}(Y)\right).} Tensor products of Hilbert spaces arise often in quantum mechanics.
|
Tensor product of Hilbert spaces
| 0.776512
|
99,942
|
The Hilbert tensor product H 1 ⊗ H 2 {\displaystyle H_{1}\otimes H_{2}} is characterized by the following universal property (Kadison & Ringrose 1997, Theorem 2.6.4): A weakly Hilbert-Schmidt mapping L: H 1 × H 2 → K {\displaystyle L:H_{1}\times H_{2}\to K} is defined as a bilinear map for which a real number d {\displaystyle d} exists, such that for all u ∈ K {\displaystyle u\in K} and one (hence all) orthonormal bases e 1 , e 2 , … {\displaystyle e_{1},e_{2},\ldots } of H 1 {\displaystyle H_{1}} and f 1 , f 2 , … {\displaystyle f_{1},f_{2},\ldots } of H 2 . {\displaystyle H_{2}.} As with any universal property, this characterizes the tensor product H uniquely, up to isomorphism. The same universal property, with obvious modifications, also applies for the tensor product of any finite number of Hilbert spaces. It is essentially the same universal property shared by all definitions of tensor products, irrespective of the spaces being tensored: this implies that any space with a tensor product is a symmetric monoidal category, and Hilbert spaces are a particular example thereof.
|
Tensor product of Hilbert spaces
| 0.776512
|
99,943
|
Since Hilbert spaces have inner products, one would like to introduce an inner product, and therefore a topology, on the tensor product that arises naturally from those of the factors. Let H 1 {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}} be two Hilbert spaces with inner products ⟨ ⋅ , ⋅ ⟩ 1 {\displaystyle \langle \cdot ,\cdot \rangle _{1}} and ⟨ ⋅ , ⋅ ⟩ 2 , {\displaystyle \langle \cdot ,\cdot \rangle _{2},} respectively. Construct the tensor product of H 1 {\displaystyle H_{1}} and H 2 {\displaystyle H_{2}} as vector spaces as explained in the article on tensor products.
|
Tensor product of Hilbert spaces
| 0.776512
|
99,944
|
This is exactly equal to the von Neumann algebra of bounded operators of H 1 ⊗ H 2 . {\displaystyle H_{1}\otimes H_{2}.} Unlike for Hilbert spaces, one may take infinite tensor products of von Neumann algebras, and for that matter C*-algebras of operators, without defining reference states. This is one advantage of the "algebraic" method in quantum statistical mechanics.
|
Tensor product of Hilbert spaces
| 0.776512
|
99,945
|
Let A i {\displaystyle {\mathfrak {A}}_{i}} be the von Neumann algebra of bounded operators on H i {\displaystyle H_{i}} for i = 1 , 2. {\displaystyle i=1,2.} Then the von Neumann tensor product of the von Neumann algebras is the strong completion of the set of all finite linear combinations of simple tensor products A 1 ⊗ A 2 {\displaystyle A_{1}\otimes A_{2}} where A i ∈ A i {\displaystyle A_{i}\in {\mathfrak {A}}_{i}} for i = 1 , 2. {\displaystyle i=1,2.}
|
Tensor product of Hilbert spaces
| 0.776512
|
99,946
|
Julian Steward's cultural ecology is considered a fusion of environmental determinism and Boas' historical approach. Steward felt it was neither nature nor culture that had the most impact on a population, but instead, the mode of subsistence used in a given environment. Anthropologist Roy Rappaport introduced the field of ecological anthropology in a deliberate attempt to move away from cultural ecology.
|
Historical ecology
| 0.776512
|
99,947
|
Historical ecology is interdisciplinary in principle; at the same time, it borrows heavily from the rich intellectual history of environmental anthropology. Western scholars have known since the time of Plato that the history of environmental changes cannot be separated from human history. Several ideas have been used to describe human interaction with the environment, the first of which is the concept of the Great Chain of Being, or inherent design in nature. In this, all forms of life are ordered, with Humanity as the highest being, due to its knowledge and ability to modify nature.
|
Historical ecology
| 0.776512
|
99,948
|
Historical ecologists recognize that humans have had world-wide influences, impact landscape in dissimilar ways which increase or decrease species diversity, and that a holistic perspective is critical to be able to understand that system.Piecing together landscapes requires a sometimes difficult union between natural and social sciences, close attention to geographic and temporal scales, a knowledge of the range of human ecological complexity, and the presentation of findings in a way that is useful to researchers in many fields. Those tasks require theory and methods drawn from geography, biology, ecology, history, sociology, anthropology, and other disciplines. Common methods include historical research, climatological reconstructions, plant and animal surveys, archaeological excavations, ethnographic interviews, and landscape reconstructions.
|
Historical ecology
| 0.776512
|
99,949
|
Historical ecology is a research program that focuses on the interactions between humans and their environment over long-term periods of time, typically over the course of centuries. In order to carry out this work, historical ecologists synthesize long-series data collected by practitioners in diverse fields. Rather than concentrating on one specific event, historical ecology aims to study and understand this interaction across both time and space in order to gain a full understanding of its cumulative effects. Through this interplay, humans adapt to and shape the environment, continuously contributing to landscape transformation.
|
Historical ecology
| 0.776512
|
99,950
|
The greater volume of people compelled the government to put pressure on farmers for increased agricultural production. Land managers no longer automatically eliminate the use of swidden fires, but rather the number of swidden fires that are set for government-sponsored agricultural purposes. The San Francisco Estuary Institute also uses historical ecology to study human impacts on the California landscape to guide environmental management.
|
Historical ecology
| 0.776512
|
99,951
|
Historical ecology can use archaeological sites within this setting to study past successes and failures of indigenous peoples. The use of swidden fires in Laos is an example of historical ecology as used by current land managers in policy-making. Swidden fires were originally considered a source of habitat degradation.
|
Historical ecology
| 0.776512
|
99,952
|
This led to an increase in the frequency and strength of wildfires, thus initiating a need to develop proper prevention methods. Historical ecology focuses on the impact on landscapes through human-mediated disturbances, once such being anthropogenic fire. It is a fusion of ecological, geographical, anthropological, and pedological interests.
|
Historical ecology
| 0.776512
|
99,953
|
Anthropogenic fire is a mechanism of human-mediated disturbance, defined within historical ecology as a means of altering the landscape in a way that better suits human needs. The most common form of anthropogenic fire is controlled burns, or broadcast burning, which people have employed for thousands of years. Forest fires and burning tend to carry negative connotations, yet controlled burns can have a favorable impact on landscape diversity, formation, and protection. Broadcast burning alters the biota of a landscape.
|
Historical ecology
| 0.776512
|
99,954
|
reef in the world. Monterey Bay National Marine Sanctuary Historical Ecology (2008–present) seeks to collect relevant historical data on fishing, whaling, and trade of the furs of aquatic animals in order form a baseline for environmental restorations of the California, USA coast.
|
Historical ecology
| 0.776512
|
99,955
|
For example, Deevey used radiocarbon dating to reconcile biologists’ successions of plants and animals with the sequences of material culture and sites discovered by archaeologists.In the 1980s, members of the history department at the University of Arkansas at Little Rock organized a lecture series entitled "Historical Ecology: Essays on Environment and Social Change" The authors noted the public's concerns with pollution and dwindling natural resources, and they began a dialogue between researchers with specialties which spanned the social sciences. The papers highlighted the importance of understanding social and political structures, personal identities, perceptions of nature, and the multiplicity of solutions for environmental problems.The emergence of historical ecology as a coherent discipline was driven by a number of long-term research projects in historical ecology of tropical, temperate and arctic environments: E.S. Deevey's Historical Ecology of the Maya Project (1973-1984) was carried out by archaeologists and biologists who combined data from lake sediments, settlement patterns, and material from excavations in the central Petén District of Guatemala to refute the hypotheses that a collapse of Mayan urban areas was instigated by faltering food production.Carole L. Crumley's Burgundian Landscape Project (1974–present) is carried out by a multidisciplinary research team aimed at identifying the multiple factors which have contributed to the long-term durability of the agricultural economy of Burgundy, France.Thomas H. McGovern's Inuit-Norse Project (1976–present) uses archaeology, environmental reconstruction, and textual analysis to examine the changing ecology of Nordic colonizers and indigenous peoples in Greenland, Iceland, Faeroes, and Shetland.In recent years the approaches to historical ecology have been expanded to include coastal and marine environments: Stellwagen Bank National Marine Sanctuary Project (1984–present) examines Massachusetts, USA cod fishing in the 17th through 19th centuries through historical records.Florida Keys Coral Reef Eco-region Project (1990–present) researchers at the Scripps Institute of Oceanography are examining archival records including natural history descriptions, maps and charts, family and personal papers, and state and colonial records in order to understand the impact of over-fishing and habitat loss in the Florida Keys, USA which contains the third largest coral reef in the world. Monterey Bay National Marine Sanctuary Historical Ecology (2008–present) seeks to collect relevant historical data on fishing, whaling, and trade of the furs of aquatic animals in order form a baseline for environmental restorations of the
|
Historical ecology
| 0.776512
|
99,956
|
The discipline has several sites of origins by researchers who shared a common interest in the problem of ecology and history, but with a diversity of approaches. Edward Smith Deevey, Jr. used the term in the 1960s to describe a methodology that had been in long development. Deevey wished to bring together the practices of "general ecology" which was studied in an experimental laboratory, with a "historical ecology" which relied on evidence collected through fieldwork.
|
Historical ecology
| 0.776512
|
99,957
|
Soil management, or direct human interaction with the soil, is another mechanism of anthropogenic change studied by historical ecologists. Soil management can take place through rearranging soils, altering drainage patterns, and building large earthen formations. Consistent with the basic premises of historical ecology, it is recognized that anthropogenic soil management practices can have both positive and negative effects on local biodiversity. Some agricultural practices have led to organically and chemically impoverished soils.
|
Historical ecology
| 0.776512
|
99,958
|
Humans have always transformed the landscapes they inhabit, however, and today there are no landscapes on Earth that have not been affected by humans in some way.Human alterations have occurred in different phases, including the period prior to industrialization. These changes have been studied through the archeological record of modern humans and their history. The evidence that classless societies, like foragers and trekkers, were able to change a landscape was a breakthrough in historical ecology and anthropology as a whole. Using an approach that combines history, ecology, and anthropology, a landscape's history can be observed and deduced through the traces of the various mechanisms that have altered it, anthropogenic or otherwise. Understanding the unique nature of every landscape, in addition to relations among landscapes, and the forms which comprise the landscape, is key to understanding historical ecology.
|
Historical ecology
| 0.776512
|
99,959
|
There are many cases in which human-mediated disturbance increases biodiversity as landscapes transform over time. Historical ecology challenges the very notion of a pristine landscape, such as virgin rainforests.
|
Historical ecology
| 0.776512
|
99,960
|
Historical ecologists recognize that landscapes undergo continuous alteration over time and these modifications are part of that landscape's history. Historical ecology recognizes that there is a primary and a secondary succession that occurs in the landscape. These successions should be understood without a preconceived bias against humanity.
|
Historical ecology
| 0.776512
|
99,961
|
These changes are due to multiple factors that contribute to the ever-changing landscape. Landscape ecology still focuses on areas defined as ecosystems. In this, the ecosystem perpetually returns to a state of equilibrium.
|
Historical ecology
| 0.776512
|
99,962
|
Sauer's pioneering 1925 paper "The Morphology of Landscape" is now fundamental to many disciplines and defines the domain. In this, the term landscape is used in a geographical sense to mean an arbitrarily selected section of reality; morphology means the conceptual and methodological processes for altering it. Hence to Sauer, wherever humans lived and impacted the environment, landscapes with determinate histories resulted.The perception of the landscape in historical ecology differs from other disciplines, such as landscape ecology.
|
Historical ecology
| 0.776512
|
99,963
|
Landscapes have cultural meanings, for example, the sacredness in many cultures of burial grounds. This recognition of landscapes as forms of knowledge is central to historical ecology, which studies landscapes from an anthropocentric perspective.The idea of the cultural landscape is directly attributed to American geographer Carl Sauer. Sauer's theories developed as a critique of environmental determinism, which was a popular theory in the early twentieth century.
|
Historical ecology
| 0.776512
|
99,964
|
While the ecosystem concept views the environment as always trying to return to a state of equilibrium, the landscape concept considers "landscape transformation" to be a process of evolution. Landscapes do not return to a state of equilibrium, but are palimpsests of successive disturbances over time. The use of "landscape" instead of "ecosystem" as the core unit of analysis lies at the heart of historical ecology.
|
Historical ecology
| 0.776512
|
99,965
|
In historical ecology, the landscape is defined as an area of interaction between human culture and the non-human environment. The landscape is a perpetually changing, physical manifestation of history. Historical ecology revises the notion of the ecosystem and replaces it with the landscape. While an ecosystem is static and cyclic, a landscape is historical.
|
Historical ecology
| 0.776512
|
99,966
|
These revisions and related critiques of environmental anthropology undertook to take into account the temporal and spatial dimensions of history and cultures, rather than continuing to view populations as static. These critiques led to the development of historical ecology by revealing the need to consider the historical, cultural, and evolutionary nature of landscapes and societies. Thus, historical ecology as a research program developed to allow for the examination of all types of societies, simple or complex, and their interactions with the environment over space and time.
|
Historical ecology
| 0.776512
|
99,967
|
Studies in ecological anthropology borrow heavily from the natural sciences, in particular, the concept of the ecosystem from systems ecology. In this approach, also called systems theory, ecosystems are seen as self-regulating, and as returning to a state of equilibrium. This theory views human populations as static and as acting in harmony with the environment.The revisions of anthropologist Eric Wolf and others are especially pertinent to the development of historical ecology.
|
Historical ecology
| 0.776512
|
99,968
|
The interdisciplinary nature of historical ecology would permit conservation biologists to create more effective and efficient landscape improvements. Reclamation and revegetation can use a historical perspective to determine what biota will be able to sustain large populations without threatening native biota of the landscape. A tropical forest in particular needs to be studied extensively because it is a highly diverse, heterogeneous setting.
|
Historical ecology
| 0.776512
|
99,969
|
Replacement of an ecosystem would create an entirely new one. Revegetation involves new additions of biota into a landscape, not limited to the original inhabitants of an area. Each method can be enriched by the application of historical ecology and the past knowledge it supplies.
|
Historical ecology
| 0.776512
|
99,970
|
Restoration ecology restores sites to former function, structure, and components of biological diversity through active modification of the landscapes. Reclamation deals with shifting a degraded ecosystem back toward a higher value or use, but not necessarily to its original state.
|
Historical ecology
| 0.776512
|
99,971
|
Historical ecology can also aid in the goals of other fields of study. Conservation biology recognizes different types of land management processes, each attempting to maintain the landscape and biota in their present form.
|
Historical ecology
| 0.776512
|
99,972
|
Through experience, many indigenous societies learned how to effectively alter their landscapes and biotic distributions. Modern societies, seeking to curtail the magnitude of their effects on the landscape, can use historical ecology to promote sustainability by learning from the past. Farmers in the Amazon region, for example, now utilize nutrient-rich terra preta to increase crop yields much like the indigenous societies that lived long before them.
|
Historical ecology
| 0.776512
|
99,973
|
Historical ecology involves an understanding of multiple fields of study such as archaeology and cultural history as well as ecological processes, species diversity, natural variability, and the impact of human-mediated disturbances. Having a broad understanding of landscapes allows historical ecology to be applied to various disciplines. Studying past relationships between humans and landscapes can successfully aid land managers by helping develop holistic, environmentally rational, and historically accurate plans of action. As summarized in the postulates of historical ecology, humans play significant roles in the creation and destruction of landscapes as well as in ecosystem function.
|
Historical ecology
| 0.776512
|
99,974
|
Historical ecologists postulate that landscape transformations have occurred throughout history, even before the dawn of western civilization. Human-mediated disturbances are predated by soil erosion and animals damming waterways which contributed to waterway transformations. Landscapes, in turn, were altered by waterway transformation. Historical ecology views the effects of human-mediated disturbances on waterway transformation as both subtle and drastic occurrences.
|
Historical ecology
| 0.776512
|
99,975
|
One example of an invasive species that has had a significant impact on the landscape is the gypsy moth (Lymantria dispar). The foliage-feeding gypsy moth is originally from temperate Eurasia; it was intentionally brought to the United States by an entomologist in 1869. Many specimens escaped from captivity and have since changed the ecology of deciduous and coniferous forests in North America by defoliation.
|
Historical ecology
| 0.776512
|
99,976
|
In physics, quantum acoustics is the study of sound under conditions such that quantum mechanical effects are relevant. For most applications, classical mechanics are sufficient to accurately describe the physics of sound. However very high frequency sounds, or sounds made at very low temperatures may be subject to quantum effects.
|
Quantum acoustics
| 0.776512
|
99,977
|
This indicates that by correctly influencing these pathways, it is theoretically possible to alter lifespan and healthspan in humans. 2017 AgeX Therapeutics, a biotechnology company focused on medical therapeutics related to longevity, was founded. 2018 The Nobel Prize for cancer research was awarded to James Allison and Tasuku Honjo.
|
Timeline of aging research
| 0.776509
|
99,978
|
2007 First evidence that a pharmacological agent (namely, metformin) at a certain dosage is capable to increase the lifespan of mice. 2008 Foundation of the Max Planck Institute for Biology of Ageing. 2008 (approximately) It was observed that different variants of FOXO3 gene are associated with human longevity.
|
Timeline of aging research
| 0.776509
|
99,979
|
Among the most notable researchers who worked in this direction, there were Alexis Carrel (who developed the technology of anastomosis of blood vessels and advanced asepsis, a Nobel laureate of 1912), Mathieu Jaboulay, Emerich Ullmann, Jacques Loeb, John Northrop, Porfiry Bakhmetiev. And although such interventions were later found to be ineffective for their intended purposes, those works led to the creation of tissue engineering, techniques for cardiopulmonary bypass and dialysis, established the foundation for the technologies for storing organs extracted from a person outside the body (which now are used, for example, during organ donation), the emergence of cryobiology. 1920s–1930s In medical practice, sex gland transplants were introduced to obtain rejuvenating effects.
|
Timeline of aging research
| 0.776509
|
99,980
|
1998 Scientists managed to extend, in a laboratory environment, the life of normal human cells beyond the Hayflick limit using telomerase. 1999 Establishment of the Buck Institute for Research on Aging – the first institute originally established primarily to study intervention into the aging process. 1999 Sierra Sciences, a biotechnology company focused on aging research with the goal of curing human aging, was founded by William H. Andrews.
|
Timeline of aging research
| 0.776509
|
99,981
|
The discovery was a revolutionary breakthrough in aging research, demonstrating that the aging process could be controlled in the laboratory, and sparked more research into the molecular biology of aging. 1995 Method for detection of senescent cells using a cytochemical assay. 1997 The absolute record for the duration of human life.
|
Timeline of aging research
| 0.776509
|
99,982
|
During centuries alchemy gradually transformed to chemistry, in parallel giving birth to many adjacent sciences or enriching them. It is worth noticing the direction of iatrochemistry – a rational direction of alchemy with the main goal of preparing medicinal products. The pioneers of iatrochemistry were Paracelsus (1493–1541), Jan Baptist van Helmont (1580–1644) and Franciscus Sylvius (1614–1672).
|
Timeline of aging research
| 0.776509
|
99,983
|
A study published a few days later identifies circadian-regulated autophagy as a critical contributor to intermittent time-restricted fasting-mediated lifespan extension in Drosophila and suggests that only certain forms of and/or combinations with intermittent fasting – intervals during which no food but only e.g. water and tea/coffee are ingested – may be effective beyond the benefits of healthy body weight. Scientists show that and how the flavonoid Procyanidin C1 of the antioxidant grape seed extract increases the health- and lifespan of mice. A vaccine to remove senescent cells, a key driver of the aging process, is demonstrated in mice by researchers from Japan. The American biotechnology company Altos Labs, which focuses on life extension research, is founded.
|
Timeline of aging research
| 0.776509
|
99,984
|
Ora Biomedical announces the "million molecule challenge", an effort to assess 1 million potential longevity interventions within five years using artificial intelligence. A study suggests chemical alternatives to age reversal via Yamanaka factors gene therapy are feasible via early in vitro fibroblasts data. These results have not yet been validated in an animal and via more reliable "improvements in age-related health metrics or lifespan".
|
Timeline of aging research
| 0.776509
|
99,985
|
Chemical intervention-related results First senolytics discovered using artificial intelligence: Teams from the University of Edinburgh and the Massachusetts Institute of Technology independently report the discovery of senolytics using artificial intelligence for screening large chemical libraries. The works reported compounds of comparable efficacy and increased potency than other known senolytics. A study identifies low levels of taurine, which declines with age, as a driver of the aging process and suggests that taurine supplements may increase lifespan.
|
Timeline of aging research
| 0.776509
|
99,986
|
In a preprint, another team of researchers of the biotechnology company Rejuvenate Bio also reports the use of Yamanaka-reprogramming to modestly extend the lives of elderly mice. However, if it was also applicable to humans, risks may include the formation of cancer. In July, the David Sinclair team at Harvard Medical School release a study that claims to have discovered the first known chemical approach to reprogram cells to a younger state by delivering the Yamanaka factors directly, whereas previously this had only been achievable via gene therapy.
|
Timeline of aging research
| 0.776509
|
99,987
|
Nutrition-related results A study reports results of the first longevity caloric restriction (CR) trial, CALERIE, finding that two years of nonintermittent CR slowed the pace of aging as measured by one of three aging clocks (modest DunedinPACE effects). Development and application of aging clocks and combination therapies A study reports the development of deep learning software using anatomic magnetic resonance images to estimate brain age with the highest accuracy for AI so far, including detecting early signs of Alzheimer's disease and varying neuroanatomical patterns of neurological aging. A study shows DNA methylation aging clocks could be useful indicators of health while social factors – such as health behaviors and poverty – are at least as good predictors and e.g. can better predict cognitive functioning. Around February, Bryan Johnson's Project Blueprint for one of the first comprehensive, possibly largely public, self-experimentations of a comprehensive combination therapy informed by the large scientific corpus on the topic and organ measurements to maximally reverse biological age and (epigenetic) aging markers achieves substantial media attention, with such activities previously largely reserved to biohackers without resources and means to evaluate effects.
|
Timeline of aging research
| 0.776509
|
99,988
|
SOCl2 adopts a trigonal pyramidal molecular geometry with Cs molecular symmetry. This geometry is attributed to the effects of the lone pair on the central sulfur(IV) center. In the solid state SOCl2 forms monoclinic crystals with the space group P21/c.
|
Cl2OS
| 0.776508
|
99,989
|
Later, Peter Gustav Lejeune Dirichlet and Bernhard Riemann expressed Fourier's results with greater precision and formality. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids. The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, shell theory, etc.
|
Fourier Series
| 0.776506
|
99,990
|
Vijay Kumar (born 12 April 1962) is an Indian roboticist and UPS foundation professor in the School of Engineering & Applied Science with secondary appointments in computer and information science and electrical and systems engineering at the University of Pennsylvania, and became the new Dean of Penn Engineering on 1 July 2015.Kumar is known for his research in the control and coordination of multi-robot formations. He was elected to the American Philosophical Society in 2018.
|
Vijay Kumar (roboticist)
| 0.776505
|
99,991
|
B.Tech., Mechanical Engineering, Indian Institute of Technology, Kanpur, India, May 1983 M.Sc., Mechanical Engineering, Ohio State University, Columbus, Ohio, March 1988 Ph.D., Mechanical Engineering, Ohio State University, Columbus, Ohio, September 1987
|
Vijay Kumar (roboticist)
| 0.776505
|
99,992
|
The Ohio State University Presidential Fellowship (1986) NSF Presidential Young Investigator Award (1991) Lindback Award for Distinguished Teaching, University of Pennsylvania (1996) The Ferdinand Freudenstein Award for significant contributions to mechanisms and robotics awarded at the 5th National Conference on Mechanisms and Robotics (1997) Best paper award, Distributed Autonomous Robotic Systems (2002) Fellow, American Society of Mechanical Engineers (2003) Kayamori Best Paper Award, IEEE International Conference on Robotics and Automation (2004) IEEE Robotics and Automation Society Distinguished Lecturer (2005) Fellow, Institute for Electrical and Electronics Engineers (2005) IEEE Robotics and Automation Society Distinguished Award (2012) George H. Heilmeier Faculty Award for Excellence in Research (2013) Member, National Academy of Engineering (2013) Popular Mechanics Breakthrough Award (2013) IIT Kanpur Distinguished Alumnus Award 2013–14, for his outstanding contributions to the area of control and coordination of multi-robot formations. The Joseph Engelberger Award by the Robotics Industries Association (2014) IEEE Robotics and Automation Award (2020)
|
Vijay Kumar (roboticist)
| 0.776505
|
99,993
|
One of the applications of polyimines is as in covalent adaptable networks (CANs). These are polymer materials that are crosslinked via dynamic covalent bonds. Besides polyimines, other types of dynamic covalent chemistry can also be used. Polyimine CANs are largely investigated to create recyclable and self-healing thermoset materials, but they can also find use in composite materials with higher performance.Flame retardants Because of the free radical scavanging properties of imines, they are well fit to be used in flame retardant materials.
|
Polyimine
| 0.776505
|
99,994
|
P. Kiss; F. Mátyás (1989). "An asymptotic formula for". Journal of Number Theory.
|
Péter Kiss (mathematician)
| 0.776504
|
99,995
|
He was born in Nagyréde, Hungary, in 1937. He majored in Mathematics and Physics from Eötvös Loránd University. After graduation, he taught mathematics at Gárdonyi Géza Secondary School in Eger. In 1971 he was appointed to Teacher's College, and in 1972 he began teaching at the Department of Mathematics of Eszterházy Károly University.He earned the Doctorate of Mathematics degree from the Hungarian Academy of Sciences in 1999.
|
Péter Kiss (mathematician)
| 0.776504
|
99,996
|
Péter Kiss ((1937-03-05)March 5, 1937 – (2002-03-05)March 5, 2002) was a Hungarian mathematician, Doctor of Mathematics, and professor of mathematics at Eszterházy Károly College, who specialized in number theory. In 1992 he won the Albert Szent-Györgyi Prize for his achievements.
|
Péter Kiss (mathematician)
| 0.776504
|
99,997
|
The MacLean Center worked for two years with transplant experts at the University of Chicago to review the ethical issues, publish protocols, and encourage professional discussion of the procedure before it was first performed on a patient. The MacLean Center’s transplant work is one example of the wide range of ethics-related research projects undertaken by its faculty. The MacLean Center faculty publishes on subjects such as research ethics, health policy, health disparities, end of life care, surgical ethics, pediatric outcomes, genetics, and transplantation ethics. Current and former MacLean Center faculty and fellows have published more than 150 books in the field of medical ethics.
|
MacLean Center for Clinical Medical Ethics
| 0.776502
|
99,998
|
Lefschetz presented his fixed-point theorem in (Lefschetz 1926). Lefschetz's focus was not on fixed points of maps, but rather on what are now called coincidence points of maps. Given two maps f {\displaystyle f} and g {\displaystyle g} from an orientable manifold X {\displaystyle X} to an orientable manifold Y {\displaystyle Y} of the same dimension, the Lefschetz coincidence number of f {\displaystyle f} and g {\displaystyle g} is defined as Λ f , g = ∑ ( − 1 ) k t r ( D X ∘ g ∗ ∘ D Y − 1 ∘ f ∗ ) , {\displaystyle \Lambda _{f,g}=\sum (-1)^{k}\mathrm {tr} (D_{X}\circ g^{*}\circ D_{Y}^{-1}\circ f_{*}),} where f ∗ {\displaystyle f_{*}} is as above, g ∗ {\displaystyle g_{*}} is the homomorphism induced by g {\displaystyle g} on the cohomology groups with rational coefficients, and D X {\displaystyle D_{X}} and D Y {\displaystyle D_{Y}} are the Poincaré duality isomorphisms for X {\displaystyle X} and Y {\displaystyle Y} , respectively. Lefschetz proved that if the coincidence number is nonzero, then f {\displaystyle f} and g {\displaystyle g} have a coincidence point. He noted in his paper that letting X = Y {\displaystyle X=Y} and letting g {\displaystyle g} be the identity map gives a simpler result, which we now know as the fixed-point theorem.
|
Lefschetz fixed-point formula
| 0.776502
|
99,999
|
For a formal statement of the theorem, let f: X → X {\displaystyle f\colon X\rightarrow X\,} be a continuous map from a compact triangulable space X {\displaystyle X} to itself. Define the Lefschetz number Λ f {\displaystyle \Lambda _{f}} of f {\displaystyle f} by Λ f := ∑ k ≥ 0 ( − 1 ) k t r ( f ∗ | H k ( X , Q ) ) , {\displaystyle \Lambda _{f}:=\sum _{k\geq 0}(-1)^{k}\mathrm {tr} (f_{*}|H_{k}(X,\mathbb {Q} )),} the alternating (finite) sum of the matrix traces of the linear maps induced by f {\displaystyle f} on H k ( X , Q ) {\displaystyle H_{k}(X,\mathbb {Q} )} , the singular homology groups of X {\displaystyle X} with rational coefficients. A simple version of the Lefschetz fixed-point theorem states: if Λ f ≠ 0 {\displaystyle \Lambda _{f}\neq 0\,} then f {\displaystyle f} has at least one fixed point, i.e., there exists at least one x {\displaystyle x} in X {\displaystyle X} such that f ( x ) = x {\displaystyle f(x)=x} . In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map homotopic to f {\displaystyle f} has a fixed point as well. Note however that the converse is not true in general: Λ f {\displaystyle \Lambda _{f}} may be zero even if f {\displaystyle f} has fixed points, as is the case for the identity map on odd-dimensional spheres.
|
Lefschetz fixed-point formula
| 0.776502
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.