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28,100 | It is a "matter of life and death" to metaphysics and to human reason, Kant argues, that the grounds of this kind of knowledge be explained.Though it received little attention when it was first published, the Critique later attracted attacks from both empiricist and rationalist critics, and became a source of controver... | Transcendental Logic | 0.806245 |
28,101 | Kant, however, argues that our knowledge of mathematics, of the first principles of natural science, and of metaphysics, is both a priori and synthetic. The peculiar nature of this knowledge cries out for explanation. The central problem of the Critique is therefore to answer the question: "How are synthetic a priori j... | Transcendental Logic | 0.806245 |
28,102 | The Critique of Pure Reason (German: Kritik der reinen Vernunft; 1781; second edition 1787) is a book by the German philosopher Immanuel Kant, in which the author seeks to determine the limits and scope of metaphysics. Also referred to as Kant's "First Critique", it was followed by his Critique of Practical Reason (178... | Transcendental Logic | 0.806245 |
28,103 | Some intuitions require the presence of their object, i.e. of the thing represented by the intuition. Other intuitions do not. (The best source for these distinctions is Kant's Lectures on Metaphysics.) | Transcendental Logic | 0.806245 |
28,104 | All philosophical concepts must be ultimately based on a posteriori, experienced intuition. This is different from algebra and geometry, which use concepts that are derived from a priori intuitions, such as symbolic equations and spatial figures. Kant's basic intention in this section of the text is to describe why rea... | Transcendental Logic | 0.806245 |
28,105 | For a one dimensional chain this result could also be reproduced using theory on aliasing. The Nyquist–Shannon sampling theorem is used in the following derivation; the main difference being that in the following derivation the discretization is not in time, but in space. If we use the correct dispersion relation from ... | Debye frequency | 0.806245 |
28,106 | Similar to a vertical screen scroll centrifuge, a horizontal screen scroll centrifuge is constructed of several main parts: screen, scroll, basket, housing, and helical screw. The screen and the basket with frustoconical geometry are assembled into the housing in a horizontal axis. Inside the frustoconical structure th... | Screen scroll centrifuge | 0.806244 |
28,107 | Governments and military groups use text mining for national security and intelligence purposes. Scientific researchers incorporate text mining approaches into efforts to organize large sets of text data (i.e., addressing the problem of unstructured data), to determine ideas communicated through text (e.g., sentiment a... | Text mining | 0.806242 |
28,108 | Subtasks—components of a larger text-analytics effort—typically include: Dimensionality reduction is important technique for pre-processing data. Technique is used to identify the root word for actual words and reduce the size of the text data. Information retrieval or identification of a corpus is a preparatory step: ... | Text mining | 0.806242 |
28,109 | The automatic analysis of vast textual corpora has created the possibility for scholars to analyze millions of documents in multiple languages with very limited manual intervention. Key enabling technologies have been parsing, machine translation, topic categorization, and machine learning. The automatic parsing of tex... | Text mining | 0.806242 |
28,110 | In experimental science, a theoretical model of reality is used. Particle physics conventionally uses a standard of "5 sigma" for the declaration of a discovery. A five-sigma level translates to one chance in 3.5 million that a random fluctuation would yield the result. This level of certainty was required in order to ... | Standard deviation | 0.806238 |
28,111 | It is algebraically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same unit as the data. The standard deviation of a population or sample and the standard error of a statistic (e.g., of the s... | Standard deviation | 0.806238 |
28,112 | In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wid... | Standard deviation | 0.806238 |
28,113 | A major step was the work of Heinrich Heesch, who first rigorously established the concept of antisymmetry as part of a series of papers in 1929 and 1930. Applying this antisymmetry operation to the 32 crystallographic point groups gives a total of 122 magnetic point groups. However, although Heesch correctly laid out ... | Magnetic space group | 0.806238 |
28,114 | In solid state physics, the magnetic space groups, or Shubnikov groups, are the symmetry groups which classify the symmetries of a crystal both in space, and in a two-valued property such as electron spin. To represent such a property, each lattice point is colored black or white, and in addition to the usual three-dim... | Magnetic space group | 0.806238 |
28,115 | Retrograde signaling in biology is the process where a signal travels backwards from a target source to its original source. For example, the nucleus of a cell is the original source for creating signaling proteins. During retrograde signaling, instead of signals leaving the nucleus, they are sent to the nucleus. In ce... | Retrograde signaling | 0.806237 |
28,116 | The family of uniform distributions over ranges of integers (with one or both bounds unknown) has a finite-dimensional sufficient statistic, namely the triple of the sample maximum, sample minimum, and sample size, but is not an exponential family of distributions, because the support varies with the parameters. For fa... | Discrete uniform random variable | 0.806236 |
28,117 | In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n. Another way of saying "discrete uniform distribution" would be "a known, finite number of ... | Discrete uniform random variable | 0.806236 |
28,118 | In 1999 the first images of atomic orbitals were obtained from a combined electron diffraction and X-Ray diffraction study of the mineral cuprite, Cu2O, by Michael O’Keeffe and John Spence of the Department of Physics at ASU. The first MOFs were designed and synthesized by Michael O’Keeffe and Omar Yagi, who at the tim... | School of Molecular Sciences (Arizona State University) | 0.806236 |
28,119 | Jones went on to teach chemistry at Baylor University for over 30 years.In 1965 Robert Pettit was hired and began developing marine-organism research that led to the creation of anti-cancer drugs and, in 1973, what became the Cancer Research Institute. Pettit taught at ASU until his retirement in 2021.In 1967, George B... | School of Molecular Sciences (Arizona State University) | 0.806236 |
28,120 | Early chemists, such as LeRoy Eyring helped build ASU's strong science reputation; Roland K. Robins conducted cancer research as early as 1957.In 1958, Arizona State College was renamed Arizona State University. Chemistry was the first department to be approved to offer a doctoral degree. In 1960, George Boyd, the univ... | School of Molecular Sciences (Arizona State University) | 0.806236 |
28,121 | In 1954 Arizona State College was restructured into 4 colleges, which went into effect in the 1955–56 academic year: the College of Liberal Arts, the College of Education, the College of Applied Arts and Sciences, and the College of Business and Public Administration. In 1957, the Department of Chemistry first appeared... | School of Molecular Sciences (Arizona State University) | 0.806236 |
28,122 | Bateman taught chemistry classes, among other things, for forty years. He oversaw the development of the physical sciences at ASU, including new science facilities and degrees.In 1946, new majors leading to degrees were added, including Physical and Biological Science. In 1947 the State of AZ designated $525,000 for a ... | School of Molecular Sciences (Arizona State University) | 0.806236 |
28,123 | Chemistry instruction at ASU can be traced back to the early 1890s. At that time, the educational institution, a Normal School for the Territory of Arizona, “acquired...a supply of chemicals” for instructional purposes. Chemistry classes were held in Old Main during the late 1800s and into the early 1900s, taught by Fr... | School of Molecular Sciences (Arizona State University) | 0.806236 |
28,124 | The School of Molecular Sciences is an academic unit of The College of Liberal Arts and Sciences at Arizona State University (ASU). The School of Molecular Sciences (SMS) is responsible for the study and teaching of the academic disciplines of chemistry and biochemistry at ASU. | School of Molecular Sciences (Arizona State University) | 0.806236 |
28,125 | Research in the School of Molecular Sciences is organized around six themes: Materials and Nanoscience Medicine and Health Energy and Sustainability Chemistry of Biology Environmental and Biogeochemistry Fundamental Molecular Science | School of Molecular Sciences (Arizona State University) | 0.806236 |
28,126 | She was inducted into the Women in Technology International (WITI) Hall of Fame, and the Hall of Fame of ASU's College of Liberal Arts and Sciences. Jeffrey Post earn his PhD from ASU in (date needed) and serves as the Mineralogist and Curator-in-Charge of Gems and Minerals at the National Museum of Natural History, th... | School of Molecular Sciences (Arizona State University) | 0.806236 |
28,127 | He served as a member of the United States House of Representatives for the state of Arizona from 1991 to 2015. Cheryl Shavers, who grew up in South Phoenix, earned both undergraduate and PhD degrees in chemistry. She subsequently worked for Motorola, Hewlett Packard, and Intel, then became Under Secretary of Commerce ... | School of Molecular Sciences (Arizona State University) | 0.806236 |
28,128 | He was a chemist and inventor who specialized in adhesives; he is credited by 3M for creating the adhesive that is used on Post-It Notes. Ed Pastor received his Bachelor of Arts degree in chemistry from ASU. | School of Molecular Sciences (Arizona State University) | 0.806236 |
28,129 | Jesse W. Jones was the first chemistry PhD and earned one of the first four doctorate degrees awarded by Arizona State University in 1963. He is a tenured Professor of Chemistry at Baylor University since 1988 and served seven two-year terms as a State Representative of Texas. In 2012 he was inducted into the African A... | School of Molecular Sciences (Arizona State University) | 0.806236 |
28,130 | The Onsager reciprocal relations have been considered the fourth law of thermodynamics. They describe the relation between thermodynamic flows and forces in non-equilibrium thermodynamics, under the assumption that thermodynamic variables can be defined locally in a condition of local equilibrium. These relations are d... | Laws of Thermodynamics | 0.806235 |
28,131 | For example, if x ∗ = y {\displaystyle x^{*}=y} then since y ∗ = x ∗ ∗ = x {\displaystyle y^{*}=x^{**}=x} in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements. In functional analysis, a linear operator A: H → H {\displaystyle A:H\to H} on a Hilbert space is called... | Self adjoint | 0.806233 |
28,132 | In mathematics, and more specifically in abstract algebra, an element x of a *-algebra is self-adjoint if x ∗ = x {\displaystyle x^{*}=x} . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection C of elements of a star-algebra is self-adjoint if it is closed under the involu... | Self adjoint | 0.806233 |
28,133 | The IUPAC Nomenclature for Transformations is a methodology for naming a chemical reaction. Traditionally, most chemical reactions, especially in organic chemistry, are named after their inventors, the so-called name reactions, such as Knoevenagel condensation, Wittig reaction, Claisen-Schmidt condensation, Schotten–Ba... | IUPAC nomenclature for organic transformations | 0.806232 |
28,134 | Advanced organic chemistry: reactions, mechanisms, and structure. New York: Wiley. ISBN 0-471-85472-7. | IUPAC nomenclature for organic transformations | 0.806232 |
28,135 | Logic of Computable Functions (LCF) is a deductive system for computable functions proposed by Dana Scott in 1969 in a memorandum unpublished until 1993. It inspired: Logic for Computable Functions (LCF), theorem proving logic by Robin Milner. Programming Computable Functions (PCF), small theoretical programming langua... | Logic of Computable Functions | 0.806228 |
28,136 | (Earlier investigations into how classical physics might be obtained as a limit of quantum mechanics had explored the subject of imperfectly isolated systems, but the role of entanglement was not fully appreciated.) A significant portion of the effort involved in quantum computing is to avoid the deleterious effects of... | Von Neumann measurement scheme | 0.806226 |
28,137 | A quantum state for an imperfectly isolated system will generally evolve to be entangled with the quantum state for the environment. Consequently, even if the system's initial state is pure, the state at a later time, found by taking the partial trace of the joint system-environment state, will be mixed. This phenomeno... | Von Neumann measurement scheme | 0.806226 |
28,138 | A 1925 paper by Heisenberg, known in English as "Quantum theoretical re-interpretation of kinematic and mechanical relations", marked a pivotal moment in the maturation of quantum physics. Heisenberg sought to develop a theory of atomic phenomena that relied only on "observable" quantities. At the time, and in contrast... | Von Neumann measurement scheme | 0.806226 |
28,139 | ... Working in various genres and media, the Fundamental Fysiks Group grappled with Bell's theorem and quantum entanglement. ... In the process, they forced a few of their physicist peers to pay attention to the topic ... From these battles, quantum information science was born. The hippie physicists' concerted push on... | Fundamental Fysiks Group | 0.806225 |
28,140 | More than most of their generation, they sought to recapture the big-picture search for meaning that had driven their heroes—Einstein, Bohr, Heisenberg, and Schrödinger—and to smuggle that mode of doing physics back into their daily routine. Second, members of the Fundamental Fysiks Group latched onto a topic, known as... | Fundamental Fysiks Group | 0.806225 |
28,141 | Hugh Gusterson writes that several challenging ideas lie at the heart of quantum physics: that electrons behave like waves and particles; that you can know a particle's location or momentum, but not both; that observing a particle changes its behavior; and that particles appear to communicate with each other across gre... | Fundamental Fysiks Group | 0.806225 |
28,142 | The Fundamental Fysiks Group was founded in San Francisco in May 1975 by two physicists, Elizabeth Rauscher and George Weissmann, at the time both graduate students at the University of California, Berkeley. The group held informal discussions on Friday afternoons to explore the philosophical implications of quantum th... | Fundamental Fysiks Group | 0.806225 |
28,143 | Specifically, in 1981, Nick Herbert, a member of the group, proposed a scheme for sending signals faster than the speed of light using quantum entanglement. Quantum computing pioneer Asher Peres writes that the refutation of Herbert's ideas led to the development of the no-cloning theorem by William Wootters, Wojciech ... | Fundamental Fysiks Group | 0.806225 |
28,144 | However, crevice corrosion is also a significant possibility and a PREN > 40 is typically specified for seawater service.These alloys need to be manufactured and heat treated correctly to be seawater corrosion resistant to the expected level. PREN alone is not an indicator of corrosion resistance. The value should be c... | Pitting resistance equivalent number | 0.806224 |
28,145 | The Éléments de géométrie algébrique ("Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut des Hautes Études Scientifiqu... | Éléments de géométrie algébrique | 0.806219 |
28,146 | In historical terms, the development of the EGA approach set the seal on the application of sheaf theory to algebraic geometry, set in motion by Serre's basic paper FAC. It also contained the first complete exposition of the algebraic approach to differential calculus, via principal parts. The foundational unification ... | Éléments de géométrie algébrique | 0.806219 |
28,147 | The following table lays out the original and revised plan of the treatise and indicates where (in SGA or elsewhere) the topics intended for the later, unpublished chapters were treated by Grothendieck and his collaborators. In addition to the actual chapters, an extensive "Chapter 0" on various preliminaries was divid... | Éléments de géométrie algébrique | 0.806219 |
28,148 | The new preface of the second edition also includes a slightly revised plan of the complete treatise, now divided into twelve chapters. Grothendieck's EGA V which deals with Bertini type theorems is to some extent available from the Grothendieck Circle website. Monografie Matematyczne in Poland has accepted this volume... | Éléments de géométrie algébrique | 0.806219 |
28,149 | The Savage representation theorem (Savage, 1954) A preference < satisfies P1–P7 if and only if there is a finitely additive probability measure P and a function u: C → R such that for every pair of acts f and g.f < g ⇐⇒ Z Ω u(f(ω)) dP ≥ Z Ω u(g(ω)) dP *If and only if all the axioms are satisfied when can used the infor... | Expected utility theory | 0.806217 |
28,150 | 1975 – Benoit Mandelbrot coins the term "fractal" to describe the self-similarity found in the Fatou, Julia and Mandelbrot sets. Fractals become the first mathematical visualization tool extensively explored with computing. 1977 – Kenneth Appel and Wolfgang Haken prove the four colour theorem, the first theorem to be p... | Timeline of scientific computing | 0.806215 |
28,151 | This decade marks the first major strides to a modern computer, and hence the start of the modern era. Fermi's Rome physics research group (informal name I ragazzi di Via Panisperna) develop statistical algorithms based on Comte de Buffon's work, that would later become the foundation of the Monte Carlo method. See als... | Timeline of scientific computing | 0.806215 |
28,152 | 1964 – Walter Kohn, with Lu Jeu Sham and Pierre Hohenberg, instigates the development of density functional theory, for which he shares the 1998 Nobel Chemistry Prize with John Pople. This contribution is arguably the earliest work to which Nobels were given for a computer program or computational technique. First regr... | Timeline of scientific computing | 0.806215 |
28,153 | 1922 – Lewis Fry Richardson introduces numerical weather forecasting by manual calculation, using methods originally developed by Vilhelm Bjerknes as early as 1895. 1926 – Grete Hermann publishes foundational paper for computer algebra, which established the existence of algorithms (including complexity bounds) for man... | Timeline of scientific computing | 0.806215 |
28,154 | 1900 – Runge’s work followed by Martin Kutta to invent the Runge-Kutta method for approximating integration for differential equations. | Timeline of scientific computing | 0.806215 |
28,155 | The one-dimensional infinite square well of length L is a model for a one-dimensional box with the potential energy: It is a standard model-system in quantum mechanics for which the solution for a single particle is well known. Since the potential inside the box is uniform, this model is referred to as 1D uniform gas, ... | Fermi gas | 0.806215 |
28,156 | The harmonic trap potential: is a model system with many applications in modern physics. The density of states (or more accurately, the degree of degeneracy) for a given spin species is: where ω ho = ω x ω y ω z 3 {\displaystyle \omega _{\text{ho}}={\sqrt{\omega _{x}\omega _{y}\omega _{z}}}} is the harmonic oscillation... | Fermi gas | 0.806215 |
28,157 | An ideal Fermi gas or free Fermi gas is a physical model assuming a collection of non-interacting fermions in a constant potential well. Fermions are elementary or composite particles with half-integer spin, thus follow Fermi-Dirac statistics. The equivalent model for integer spin particles is called the Bose gas (an e... | Fermi gas | 0.806215 |
28,158 | Related to the Fermi energy, a few useful quantities also occur often in modern literature. The Fermi temperature is defined as T F = E F k B {\textstyle T_{\mathrm {F} }={\frac {E_{\mathrm {F} }}{k_{\rm {B}}}}} , where k B {\displaystyle k_{\rm {B}}} is the Boltzmann constant. The Fermi temperature can be thought of a... | Fermi gas | 0.806215 |
28,159 | In physics, a Dirac string is a one-dimensional curve in space, conceived of by the physicist Paul Dirac, stretching between two hypothetical Dirac monopoles with opposite magnetic charges, or from one magnetic monopole out to infinity. The gauge potential cannot be defined on the Dirac string, but it is defined everyw... | Dirac string | 0.806214 |
28,160 | If d {\displaystyle d} is a square-free integer (a product of distinct primes) other than 1, then Q ( d ) {\displaystyle \mathbf {Q} ({\sqrt {d}}\,)} is a quadratic extension of Q. If d < 0 {\displaystyle d<0} , then the class number of the ring R {\displaystyle R} of algebraic integers of Q ( d ) {\displaystyle \mathb... | Ideal class group | 0.806214 |
28,161 | The principal ideals form the ideal class which serves as an identity element for this multiplication. Thus a class has an inverse if and only if there is an ideal J such that IJ is a principal ideal. In general, such a J may not exist and consequently the set of ideal classes of R may only be a monoid. However, if ... | Ideal class group | 0.806214 |
28,162 | In number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group JK /PK where JK is the group of fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ... | Ideal class group | 0.806214 |
28,163 | This result gives a bound, depending on the ring, such that every ideal class contains an ideal norm less than the bound. In general the bound is not sharp enough to make the calculation practical for fields with large discriminant, but computers are well suited to the task. The mapping from rings of integers R to thei... | Ideal class group | 0.806214 |
28,164 | But if R is a ring of algebraic integers, then the class number is always finite. This is one of the main results of classical algebraic number theory. Computation of the class group is hard, in general; it can be done by hand for the ring of integers in an algebraic number field of small discriminant, using Minkowski'... | Ideal class group | 0.806214 |
28,165 | Somewhat later again Richard Dedekind formulated the concept of an ideal, Kummer having worked in a different way. At this point the existing examples could be unified. It was shown that while rings of algebraic integers do not always have unique factorization into primes (because they need not be principal ideal domai... | Ideal class group | 0.806214 |
28,166 | It had been realised (probably by several people) that failure to complete proofs in the general case of Fermat's Last Theorem by factorisation using the roots of unity was for a very good reason: a failure of unique factorization – i.e., the fundamental theorem of arithmetic – to hold in the rings generated by those r... | Ideal class group | 0.806214 |
28,167 | Class field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group. A particularly beautiful example is found in the Hilbert class field of a number field, which can be defined as the maximal... | Ideal class group | 0.806214 |
28,168 | In mathematics, given two submanifolds A and B of a manifold X intersecting in two points p and q, a Whitney disc is a mapping from the two-dimensional disc D, with two marked points, to X, such that the two marked points go to p and q, one boundary arc of D goes to A and the other to B.Their existence and embeddedness... | Whitney disk | 0.806212 |
28,169 | In statistics, for example in statistical quality control, a statistical assembly is a collection of parts or components which makes up a statistical unit. Thus a statistical unit, which would be the prime item of concern, is made of discrete components like organs or machine parts. The reliability of the statistical u... | Statistical assembly | 0.806211 |
28,170 | They have been used recently for truckload transportation, bus routes, industrial procurement, and in the allocation of radio spectrum for wireless communications. In recent years, procurement teams have applied reverse combinatorial auctions in the procurement of goods and services. This application is often referred ... | Combinatorial auction | 0.806206 |
28,171 | It is an open question to give support for a "Church thesis" for fuzzy mathematics, the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. In order to solve this, an extension of the notions of fuzzy grammar and fuzzy Turing machine are necessary. Another open question is to start from th... | Fuzzy logics | 0.806204 |
28,172 | An extension of such a theory to the general case of the L-subsets is possible (see Gerla 2006). The proposed definitions are well related to fuzzy logic. Indeed, the following theorem holds true (provided that the deduction apparatus of the considered fuzzy logic satisfies some obvious effectiveness property). | Fuzzy logics | 0.806204 |
28,173 | In the 1980s, researchers were divided about the most effective approach to machine learning: deductive models or neural networks. The former approach requires large decision trees and uses binary logic, matching the hardware on which it runs. The physical devices might be limited to binary logic, but AI can use softwa... | Fuzzy logics | 0.806204 |
28,174 | Neural network-based artificial intelligence and fuzzy logic, when analyzed, are the same thing—the underlying logic of neural networks is fuzzy. A neural network will take a variety of valued inputs, give them different weights in relation to each other, and arrive at a decision, which normally also has a value. Nowhe... | Fuzzy logics | 0.806204 |
28,175 | When Hatfield left JPL in 1974, Jeb Long took over his role.While working at JPL as a contractor, C. Wayne Ratliff entered the office football pool. He had no interest in the game as such, but felt he could win the pool by processing the post-game statistics found in newspapers. In order to do this, he turned his atten... | DBase II | 0.806204 |
28,176 | For handling data, dBase provided detailed procedural commands and functions to open and traverse records in data files (e.g., USE, SKIP, GO TOP, GO BOTTOM, and GO recno), manipulate field values (REPLACE and STORE), and manipulate text strings (e.g., STR() and SUBSTR()), numbers, and dates.dBase is an application deve... | DBase II | 0.806204 |
28,177 | dBase (also stylized dBASE) was one of the first database management systems for microcomputers and the most successful in its day. The dBase system includes the core database engine, a query system, a forms engine, and a programming language that ties all of these components together. dBase's underlying file format, t... | DBase II | 0.806204 |
28,178 | All equilibrium constants vary with temperature according to the van 't Hoff equation d ln ( K ) d T = Δ H ⊖ R T 2 {\displaystyle {\frac {\mathrm {d} \ln \left(K\right)}{\mathrm {d} T}}={\frac {\Delta H^{\ominus }}{RT^{2}}}} R {\displaystyle R} is the gas constant and T {\displaystyle T} is the absolute temperature. ... | Ionization constant | 0.806199 |
28,179 | In living organisms, acid–base homeostasis and enzyme kinetics are dependent on the pKa values of the many acids and bases present in the cell and in the body. In chemistry, a knowledge of pKa values is necessary for the preparation of buffer solutions and is also a prerequisite for a quantitative understanding of the ... | Ionization constant | 0.806199 |
28,180 | These calculations find application in many different areas of chemistry, biology, medicine, and geology. For example, many compounds used for medication are weak acids or bases, and a knowledge of the pKa values, together with the octanol-water partition coefficient, can be used for estimating the extent to which the ... | Ionization constant | 0.806199 |
28,181 | As a metabolomics resource, the ECMDB is designed to facilitate research in the area gut/microbiome metabolomics and environmental metabolomics. The ECMDB contains two kinds of data: 1) chemical data and 2) molecular biology and/or biochemical data. The chemical data includes more than 2700 metabolite structures with d... | E. Coli Metabolome Database | 0.806199 |
28,182 | The E. coli Metabolome Database (ECMDB) is a comprehensive, high-quality, freely accessible, online database of small molecule metabolites found in or produced by Escherichia coli (E. coli strain K12, MG1655). Escherichia coli is perhaps the best studied bacterium on earth and has served as the "model microbe" in micro... | E. Coli Metabolome Database | 0.806199 |
28,183 | In molecular biology, this protein domain belongs to the terpene synthase family (TPS). Its role is to synthesize terpenes, which are part of primary metabolism, such as sterols and carotene, and also part of the secondary metabolism. This entry will focus on the N terminal domain of the TPS protein. | Terpene synthase N terminal domain | 0.806198 |
28,184 | ), Basel–Boston–Berlin: Birkhäuser Verlag, pp. xx+476, doi:10.1007/978-3-0348-0513-1, ISBN 978-3-0348-0512-4, MR 3014456, Zbl 1270.35005. Stampacchia, Guido (1964), "Formes bilineaires coercitives sur les ensembles convexes", Comptes rendus hebdomadaires des séances de l'Académie des sciences, 258: 4413–4416, Zbl 0124.... | Variational inequalities | 0.806198 |
28,185 | Lions, Jacques-Louis; Stampacchia, Guido (1967), "Variational inequalities", Communications on Pure and Applied Mathematics, 20 (3): 493–519, doi:10.1002/cpa.3160200302, Zbl 0152.34601, archived from the original on 2013-01-05. An important paper, describing the abstract approach of the authors to the theory of variati... | Variational inequalities | 0.806198 |
28,186 | Fichera, Gaetano (1964a), "Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno" , Memorie della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 8 (in Italian), 7 (2): 91–140, Zbl 0146.21204. The first paper where an existence and... | Variational inequalities | 0.806198 |
28,187 | In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities was initially developed to deal with equilibrium problems, precisely the Signorini ... | Variational inequalities | 0.806198 |
28,188 | The normalized sinc function sinc ( x ) = sin ( π x ) π x {\displaystyle {\text{sinc}}(x)={\frac {\sin(\pi x)}{\pi x}}} has a Weierstrass factorization representation as an infinite product: The infinite product is analytic, so taking the natural logarithm of both sides and differentiating yields (by uniform converge... | Basel problem | 0.806196 |
28,189 | Given a complete orthonormal basis in the space L per 2 ( 0 , 1 ) {\displaystyle L_{\operatorname {per} }^{2}(0,1)} of L2 periodic functions over ( 0 , 1 ) {\displaystyle (0,1)} (i.e., the subspace of square-integrable functions which are also periodic), denoted by { e i } i = − ∞ ∞ {\displaystyle \{e_{i}\}_{i=-\infty ... | Basel problem | 0.806196 |
28,190 | Euler's original derivation of the value π 2 / 6 {\displaystyle \pi ^{2}/6} essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series. Of course, Euler's original reasoning requires justification (100 years later, Karl Weierstrass proved that Euler's... | Basel problem | 0.806196 |
28,191 | The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. Since the problem had withst... | Basel problem | 0.806196 |
28,192 | While most proofs use results from advanced mathematics, such as Fourier analysis, complex analysis, and multivariable calculus, the following does not even require single-variable calculus (until a single limit is taken at the end). For a proof using the residue theorem, see here. | Basel problem | 0.806196 |
28,193 | Putting these together gives ω ( S L 2 ( Z p ) ) = ( 1 − 1 / p 2 ) {\displaystyle \omega (SL_{2}(\mathbb {Z} _{p}))=(1-1/p^{2})} . At the infinite place, an integral computation over the fundamental domain of S L 2 ( Z ) {\displaystyle SL_{2}(\mathbb {Z} )} shows that ω ( S L 2 ( Z ) ∖ S L 2 ( R ) = π 2 / 6 {\displayst... | Basel problem | 0.806196 |
28,194 | A proof is also possible assuming Weil's conjecture on Tamagawa numbers. The conjecture asserts for the case of the algebraic group SL2(R) that the Tamagawa number of the group is one. That is, the quotient of the special linear group over the rational adeles by the special linear group of the rationals (a compact set,... | Basel problem | 0.806196 |
28,195 | Let x be a real number with 0 < x < π/2, and let n be a positive odd integer. Then from de Moivre's formula and the definition of the cotangent function, we have From the binomial theorem, we have Combining the two equations and equating imaginary parts gives the identity We take this identity, fix a positive integer m... | Basel problem | 0.806196 |
28,196 | Since the optical transfer function is the Fourier transform of the point spread function, and the point spread function is the square absolute of the inverse Fourier transformed pupil function, the optical transfer function can also be calculated directly from the pupil function. From the convolution theorem it can be... | Optical Transfer Function | 0.806193 |
28,197 | Thus, if the mean is intended as a measure of the location of the center of the data, it is, in a sense, biased when outliers are present. Also, the distribution of the mean is known to be asymptotically normal due to the central limit theorem. However, outliers can make the distribution of the mean non-normal even for... | Resistant statistic | 0.80619 |
28,198 | Unfortunately, when there are outliers in the data, classical estimators often have very poor performance, when judged using the breakdown point and the influence function, described below. The practical effect of problems seen in the influence function can be studied empirically by examining the sampling distribution ... | Resistant statistic | 0.80619 |
28,199 | Robust statistics seek to provide methods that emulate popular statistical methods, but are not unduly affected by outliers or other small departures from model assumptions. In statistics, classical estimation methods rely heavily on assumptions that are often not met in practice. In particular, it is often assumed tha... | Resistant statistic | 0.80619 |
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