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7,700 | The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through t... | Parallel axes rule | 0.822397 |
7,701 | Suppose a body of mass m is rotated about an axis z passing through the body's center of mass. The body has a moment of inertia Icm with respect to this axis. The parallel axis theorem states that if the body is made to rotate instead about a new axis z′, which is parallel to the first axis and displaced from it by a d... | Parallel axes rule | 0.822397 |
7,702 | By construction the disc algebra is a closed subalgebra of the Hardy space H∞. In contrast to the stronger requirement that a continuous extension to the circle exists, it is a lemma of Fatou that a general element of H∞ can be radially extended to the circle almost everywhere. == References == | Disk algebra | 0.822394 |
7,703 | In mathematics, specifically in functional and complex analysis, the disk algebra A(D) (also spelled disc algebra) is the set of holomorphic functions ƒ: D → C {\displaystyle \mathbb {C} } ,(where D is the open unit disk in the complex plane C {\displaystyle \mathbb {C} } ) that extend to a continuous function on the c... | Disk algebra | 0.822394 |
7,704 | In addition to having a diffusive effect on ion transport, water molecules also form hydration shells around individual ions due to their polar nature. The hydration shell not only shields the charge on ions from other ions but also modulates the ion radial distribution function causing the formation of peaks and troug... | Biology Monte Carlo method | 0.822393 |
7,705 | Biology Monte Carlo methods (BioMOCA) have been developed at the University of Illinois at Urbana-Champaign to simulate ion transport in an electrolyte environment through ion channels or nano-pores embedded in membranes. It is a 3-D particle-based Monte Carlo simulator for analyzing and studying the ion transport prob... | Biology Monte Carlo method | 0.822393 |
7,706 | As this approach could add a significant load to the amount of calculations, a simpler approach is chosen that treats the protein surfaces as predetermined hard wall boundaries. Many recent open source molecular biology packages have built-in facilities that determine the volume accessible to ions in a protein system. ... | Biology Monte Carlo method | 0.822393 |
7,707 | Other than counting the number of ions crossing the channel, sometimes it is desirable to study their behavior at different regions of the channel. Such examples would be the average occupancy of ions or their average moving velocity inside the channel or a nanopore. BioMOCA has been equipped with the option of dumping... | Biology Monte Carlo method | 0.822393 |
7,708 | In order to use box integration for discretizing a D-dimensional Poisson equation ∇ ( ε ∇ φ ) = ρ {\displaystyle \nabla (\varepsilon \nabla \varphi )=\rho } with ε {\displaystyle \varepsilon } being a diagonal D × D tensor, this differential equation is reformulated as an integral equation. Integration the above equati... | Biology Monte Carlo method | 0.822393 |
7,709 | The use of specialized and efficient libraries to provide more terse abstractions is also common in other programming languages. In C++ several linear algebra libraries exploit the language's ability to overload operators. In some cases a very terse abstraction in those languages is explicitly influenced by the array p... | Array programming | 0.822392 |
7,710 | If the system is overdetermined – so that A has more rows than columns – the pseudoinverse A+ (in MATLAB and GNU Octave languages: pinv(A)) can replace the inverse A−1, as follows: pinv(A) *(A * x)==pinv(A) * (b) (pinv(A) * A)* x ==pinv(A) * b (matrix-multiplication associativity) x = pinv(A) * bHowever, these solution... | Array programming | 0.822392 |
7,711 | The implementation in MATLAB allows the same economy allowed by using the Fortran language. A variant of the MATLAB language is the GNU Octave language, which extends the original language with augmented assignments: Both MATLAB and GNU Octave natively support linear algebra operations such as matrix multiplication, ma... | Array programming | 0.822392 |
7,712 | In computer science, array programming refers to solutions that allow the application of operations to an entire set of values at once. Such solutions are commonly used in scientific and engineering settings. Modern programming languages that support array programming (also known as vector or multidimensional languages... | Array programming | 0.822392 |
7,713 | In scalar languages such as C and Pascal, operations apply only to single values, so a+b expresses the addition of two numbers. In such languages, adding one array to another requires indexing and looping, the coding of which is tedious. In array-based languages, for example in Fortran, the nested for-loop above can be... | Array programming | 0.822392 |
7,714 | Another paper by Carrell and West (2010) use a data set from the U.S. Air Force Academy where students are randomly assigned to course sections (reducing selection problems). It found that calculus students got higher marks on common course examinations when they had instructors with high SET scores but did worse when ... | Course evaluation | 0.822388 |
7,715 | The hybrid QM/MM (quantum mechanics/molecular mechanics) approach is a molecular simulation method that combines the strengths of ab initio QM calculations (accuracy) and MM (speed) approaches, thus allowing for the study of chemical processes in solution and in proteins. The QM/MM approach was introduced in the 1976 p... | QM/MM | 0.822386 |
7,716 | Link atom schemes introduce an additional atomic centre (usually a hydrogen atom). This atom is not part of the real system. It is covalently bonded to the atom being described by quantum mechanics which serves to saturate its valency (by replacing the bond that has been broken). | QM/MM | 0.822386 |
7,717 | The Institute of Molecular Biology and Genetics of the National Academy of Sciences of Ukraine (IMBG of the NASU) is a scientific research organisation in Kyiv, Ukraine. Founded in 1973 as a branch of the National Academy of Sciences of Ukraine, the institute specialises in genomics, proteomics, bioinformatics, genetic... | NASU Institute of Molecular Biology and Genetics | 0.822385 |
7,718 | Latest generation transmissometer technology makes use of a co-located forward scatter visibility sensor on the transmitter unit to allow for higher accuracies over an Extended Meteorological Optical Range or EMOR. After 10,000 meters the accuracy of transmissometer technology diminishes, and at higher visibilities for... | Transmissometer | 0.822385 |
7,719 | Fundamental theorem of algebra – states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero. Equations – equality of two mathematical... | Outline of algebra | 0.82237 |
7,720 | An algebraic equation is an equation involving only algebraic expressions in the unknowns. These are further classified by degree. Linear equation – algebraic equation of degree one. | Outline of algebra | 0.82237 |
7,721 | Algebra is one of the main branches of mathematics, covering the study of structure, relation and quantity. Algebra studies the effects of adding and multiplying numbers, variables, and polynomials, along with their factorization and determining their roots. In addition to working directly with numbers, algebra also co... | Outline of algebra | 0.82237 |
7,722 | Pre-algebra Elementary algebra Boolean algebra Abstract algebra Linear algebra Universal algebra | Outline of algebra | 0.82237 |
7,723 | Finally, the two random variables X and Y are equal if they are equal as functions on their measurable space: X ( ω ) = Y ( ω ) for all ω . {\displaystyle X(\omega )=Y(\omega )\qquad {\hbox{for all }}\omega .} This notion is typically the least useful in probability theory because in practice and in theory, the underly... | Random Variable | 0.822362 |
7,724 | A significant theme in mathematical statistics consists of obtaining convergence results for certain sequences of random variables; for instance the law of large numbers and the central limit theorem. There are various senses in which a sequence X n {\displaystyle X_{n}} of random variables can converge to a random var... | Random Variable | 0.822362 |
7,725 | Other ranges of values would have half the probabilities of the last example. Most generally, every probability distribution on the real line is a mixture of discrete part, singular part, and an absolutely continuous part; see Lebesgue's decomposition theorem § Refinement. The discrete part is concentrated on a countab... | Random Variable | 0.822362 |
7,726 | UniProt NCBI, National Center for Biotechnology Information | Ancient protein | 0.822361 |
7,727 | Another common protein in the archaeological record is milk beta-lactoglobulin, often recovered from ancient dental calculi. Beta-lactoglobulin is a small whey protein with a molecular mass of around 18400 Da (dalton). It is resistant to heating and enzymatic degradation; structurally, it has a beta-barrel associated w... | Ancient protein | 0.822361 |
7,728 | Protein sequence data can be downloaded from public genebanks (UniProt/NCBI) and exported as FASTA files for sequencing algorithms. Recently, open search engines such as MetaMorpheus, pFind and Fragpipe have received attention, because they make it possible to identify all modifications associated with peptide spectral... | Ancient protein | 0.822361 |
7,729 | Jope and Wesbroek were leading experts in shell proteins and crystallisation. Wesbroek later established Geobiochemistry laboratory at the University of Leiden, focusing on biomineralisation and how this process facilitated protein survival. He also pioneered the use of antibodies for the study of ancient proteins in t... | Ancient protein | 0.822361 |
7,730 | Despite significant analytical challenges, palaeoproteomics is constantly evolving and adopting new technology. Latest high-performance mass spectrometry, for example, TimsToF (trapped ion mobility-time-of-flight) in a DIA mode (data independent acquisition) may help with the separation, selection and resolution of anc... | Ancient protein | 0.822361 |
7,731 | Palaeoproteomics is a fast-developing field that combines archaeology, biology, chemistry and heritage studies. Comparable to its high-profile sister field, aDNA analysis, the extraction, identification and authentication of ancient proteins are challenging, since both ancient DNA and proteins tend to be ultrashort, hi... | Ancient protein | 0.822361 |
7,732 | Cereal Crops Division Feed and Industrial Crops Division Fruit and Ornamental Crops Division Legume Crops Division Vegetables and Special Crops Division Plant Pathology Laboratory Entomology Laboratory Plant Physiology Laboratory Extension Division Genetics Laboratory National Plant Genetic Resources Laboratory Biochem... | University of the Philippines Los Baños Institute of Plant Breeding | 0.822354 |
7,733 | The Institute of Plant Breeding (IPB) is a research institute of the University of the Philippines Los Baños. It is the national biotechnology research center and repository for all crops other than rice, which is handled by the Philippine Rice Research Institute. It traces its roots to the Plant Breeding Division of t... | University of the Philippines Los Baños Institute of Plant Breeding | 0.822354 |
7,734 | Thus, rings can be described as the so-called "algebras" of some operad, but not groups, since the law g g − 1 = 1 {\displaystyle gg^{-1}=1} duplicates the variable g on the left side and omits it on the right side. At first this may seem to be a troublesome restriction, but the payoff is that operads have certain adva... | Equational reasoning | 0.82235 |
7,735 | In particular, every Lawvere theory gives a monad on the category of sets, while any "finitary" monad on the category of sets arises from a Lawvere theory. However, a monad describes algebraic structures within one particular category (for example the category of sets), while algebraic theories describe structure withi... | Equational reasoning | 0.82235 |
7,736 | Universal algebra has also been studied using the techniques of category theory. In this approach, instead of writing a list of operations and equations obeyed by those operations, one can describe an algebraic structure using categories of a special sort, known as Lawvere theories or more generally algebraic theories.... | Equational reasoning | 0.82235 |
7,737 | After the operations have been specified, the nature of the algebra is further defined by axioms, which in universal algebra often take the form of identities, or equational laws. An example is the associative axiom for a binary operation, which is given by the equation x ∗ (y ∗ z) = (x ∗ y) ∗ z. The axiom is intended ... | Equational reasoning | 0.82235 |
7,738 | The class of fields is not an equational class because there is no type (or "signature") in which all field laws can be written as equations (inverses of elements are defined for all non-zero elements in a field, so inversion cannot be added to the type). One advantage of this restriction is that the structures studied... | Equational reasoning | 0.82235 |
7,739 | A collection of algebraic structures defined by identities is called a variety or equational class. Restricting one's study to varieties rules out: quantification, including universal quantification ( ∀ {\displaystyle \forall } ) except before an equation, and existential quantification ( ∃ {\displaystyle \exists } ) l... | Equational reasoning | 0.82235 |
7,740 | Most of the usual algebraic systems of mathematics are examples of varieties, but not always in an obvious way, since the usual definitions often involve quantification or inequalities. | Equational reasoning | 0.82235 |
7,741 | Most algebraic structures are examples of universal algebras. Rings, semigroups, quasigroups, groupoids, magmas, loops, and others. Vector spaces over a fixed field and modules over a fixed ring are universal algebras. These have a binary addition and a family of unary scalar multiplication operators, one for each elem... | Equational reasoning | 0.82235 |
7,742 | In universal algebra, an algebra (or algebraic structure) is a set A together with a collection of operations on A. An n-ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a 0-ary operation (or nullary operation) can be represented simply as an element of A, or a consta... | Equational reasoning | 0.82235 |
7,743 | Ladder Logic was the game for the 1998 FIRST Robotics Competition. | Ladder Logic | 0.822349 |
7,744 | These organisms utilize this symbiotic relationship in order to utilize and obtain the chemical energy that is released at these hydrothermal vent areas. Chemolithoautotrophic bacteria can derive nutrients and energy from the geological activity at a hydrothermal vent to fix carbon into organic forms.Viruses are also a... | Marine microorganism | 0.822349 |
7,745 | Sea ice microbial communities (SIMCO) refer to groups of microorganisms living within and at the interfaces of sea ice at the poles. The ice matrix they inhabit has strong vertical gradients of salinity, light, temperature and nutrients. Sea ice chemistry is most influenced by the salinity of the brine which affects th... | Marine microorganism | 0.822349 |
7,746 | In 1987, Kent Beck and Ward Cunningham began experimenting with the idea of applying patterns to programming – specifically pattern languages – and presented their results at the OOPSLA conference that year. In the following years, Beck, Cunningham and others followed up on this work. Design patterns gained popularity ... | Programming pattern | 0.822345 |
7,747 | In software engineering, a software design pattern is a general, reusable solution to a commonly occurring problem within a given context in software design. It is not a finished design that can be transformed directly into source or machine code. Rather, it is a description or template for how to solve a problem that ... | Programming pattern | 0.822345 |
7,748 | Developments in the following decades include the application of statistical mechanics to chemical systems and work on colloids and surface chemistry, where Irving Langmuir made many contributions. Another important step was the development of quantum mechanics into quantum chemistry from the 1930s, where Linus Pauling... | Physical Chemistry | 0.822344 |
7,749 | This paper introduced several of the cornerstones of physical chemistry, such as Gibbs energy, chemical potentials, and Gibbs' phase rule.The first scientific journal specifically in the field of physical chemistry was the German journal, Zeitschrift für Physikalische Chemie, founded in 1887 by Wilhelm Ostwald and Jaco... | Physical Chemistry | 0.822344 |
7,750 | The term "physical chemistry" was coined by Mikhail Lomonosov in 1752, when he presented a lecture course entitled "A Course in True Physical Chemistry" (Russian: Курс истинной физической химии) before the students of Petersburg University. In the preamble to these lectures he gives the definition: "Physical chemistry ... | Physical Chemistry | 0.822344 |
7,751 | Theoretical developments have gone hand in hand with developments in experimental methods, where the use of different forms of spectroscopy, such as infrared spectroscopy, microwave spectroscopy, electron paramagnetic resonance and nuclear magnetic resonance spectroscopy, is probably the most important 20th century dev... | Physical Chemistry | 0.822344 |
7,752 | The fact that how fast reactions occur can often be specified with just a few concentrations and a temperature, instead of needing to know all the positions and speeds of every molecule in a mixture, is a special case of another key concept in physical chemistry, which is that to the extent an engineer needs to know, e... | Physical Chemistry | 0.822344 |
7,753 | However, classical thermodynamics is mostly concerned with systems in equilibrium and reversible changes and not what actually does happen, or how fast, away from equilibrium. Which reactions do occur and how fast is the subject of chemical kinetics, another branch of physical chemistry. A key idea in chemical kinetics... | Physical Chemistry | 0.822344 |
7,754 | Quantum chemistry, a subfield of physical chemistry especially concerned with the application of quantum mechanics to chemical problems, provides tools to determine how strong and what shape bonds are, how nuclei move, and how light can be absorbed or emitted by a chemical compound. Spectroscopy is the related sub-disc... | Physical Chemistry | 0.822344 |
7,755 | Reactions of electrochemical cells. Behaviour of microscopic systems using quantum mechanics and macroscopic systems using statistical thermodynamics. Calculation of the Energy of electron movement in a metal complexes. | Physical Chemistry | 0.822344 |
7,756 | Transfer of heat between a chemical system and its surroundings during change of phase or chemical reaction taking place called thermochemistry Study of colligative properties of number of species present in solution. Number of phases, number of components and degree of freedom (or variance) can be correlated with one ... | Physical Chemistry | 0.822344 |
7,757 | The identity of ions and the electrical conductivity of materials. Surface science and electrochemistry of cell membranes. Interaction of one body with another in terms of quantities of heat and work called thermodynamics. | Physical Chemistry | 0.822344 |
7,758 | Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical mechanics, analytical dynamics and chemical equilibria. Physical chemistry, in c... | Physical Chemistry | 0.822344 |
7,759 | Sedimentology and stratigraphy are used to read the rocks. The rock record stores a history of geobiologic processes in sediments which can be unearthed through an understanding of deposition, sedimentation, compaction, diagenesis, and deformation. The search for and study of fossils, while often delegated to the separ... | Geobiology | 0.822333 |
7,760 | While geobiology is a diverse and varied field, encompassing ideas and techniques from a wide range of disciplines, there are a number of important methods that are key to the study of the interaction of life and Earth that are highlighted here. Laboratory culturing of microbes is used to characterize the metabolism an... | Geobiology | 0.822333 |
7,761 | In the 1970s and '80s, scientists like Geoffrey Eglington and Roger Summons began to find lipid biomarkers in the rock record using equipment like GCMS.On the biology side of things, in 1977, Carl Woese and George Fox published a phylogeny of life on Earth, including a new domain - the Archaea. And in the 1990s, geneti... | Geobiology | 0.822333 |
7,762 | However, it took another 40 or so years for geobiology to become a firmly rooted scientific discipline, thanks in part to advances in geochemistry and genetics that enabled scientists to begin to synthesize the study of life and planet. In the 1930s, Alfred Treibs discovered chlorophyll-like porphyrins in petroleum, co... | Geobiology | 0.822333 |
7,763 | In his book, Geobiology, Bass Becking stated that he had no intention of inventing a new field of study. Baas Becking's understanding of geobiology was heavily influenced by his predecessors, including Martinus Beyerinck, his teacher from the Dutch School of Microbiology. Others included Vladimir Vernadsky, who argued ... | Geobiology | 0.822333 |
7,764 | The term geobiology was coined by Lourens Baas Becking in 1934. In his words, geobiology "is an attempt to describe the relationship between organisms and the Earth," for "the organism is part of the Earth and its lot is interwoven with that of the Earth." Baas Becking's definition of geobiology was born of a desire to... | Geobiology | 0.822333 |
7,765 | This is an important concept in the field because it represents the epitome of the interconnectedness, if not sameness, of life and Earth.While often delegated to the field of astrobiology, attempts to understand how and when life arose are relevant to geobiology as well. The first major strides towards understanding t... | Geobiology | 0.822333 |
7,766 | The first life arose from abiotic chemical reactions. When this happened, how it happened, and even what planet it happened on are uncertain. However, life follows the rules of and arose from lifeless chemistry and physics. It is constrained by principles such as thermodynamics. | Geobiology | 0.822333 |
7,767 | Specifically, it asks questions about where microbes live, their local and global abundance, their structural and functional biochemistry, how they have evolved, biomineralization, and their preservation potential and presence in the rock record. In many ways, GMG appears to be equivalent to geobiology, but differs in ... | Geobiology | 0.822333 |
7,768 | Geomicrobiology traditionally studies the interactions between microbes and minerals. While it is generally reliant on the tools of microbiology, microbial geochemistry uses geological and chemical methods to approach the same topic from the perspective of the rocks. Geomicrobiology and microbial geochemistry (GMG) is ... | Geobiology | 0.822333 |
7,769 | Evolutionary biology is the study of the evolutionary processes that have shaped the diversity of life on Earth. It incorporates genetics, ecology, biogeography, and paleontology to analyze topics including natural selection, variance, adaptation, divergence, genetic drift, and speciation. | Geobiology | 0.822333 |
7,770 | Biogeochemistry is a systems science that synthesizes the study of biological, geological, and chemical processes to understand the reactions and composition of the natural environment. It is concerned primarily with global elemental cycles, such as that of nitrogen and carbon. The father of biogeochemistry was James L... | Geobiology | 0.822333 |
7,771 | Astrobiology is an interdisciplinary field that uses a combination of geobiological and planetary science data to establish a context for the search for life on other planets. The origin of life from non-living chemistry and geology, or abiogenesis, is a major topic in astrobiology. Even though it is fundamentally an e... | Geobiology | 0.822333 |
7,772 | As its name suggests, geobiology is closely related to many other fields of study, and does not have clearly defined boundaries or perfect agreement on what exactly they comprise. Some practitioners take a very broad view of its boundaries, encompassing many older, more established fields such as biogeochemistry, paleo... | Geobiology | 0.822333 |
7,773 | A core concept in geobiology is that life changes over time through evolution. The theory of evolution postulates that unique populations of organisms or species arose from genetic modifications in the ancestral population which were passed down by drift and natural selection.Along with standard biological evolution, l... | Geobiology | 0.822333 |
7,774 | The sedimentary record allows scientists to observe changes in life and Earth in composition over time and sometimes even date major transitions, like extinction events. Some classic examples of geobiology in the sedimentary record include stromatolites and banded-iron formations. The role of life in the origin of both... | Geobiology | 0.822333 |
7,775 | Putative bacterial microfossils and ancient stromatolites are used as evidence for the rise of metabolisms such as oxygenic photosynthesis. The search for molecular fossils, such as lipid biomarkers like steranes and hopanes, has also played an important role in geobiology and organic geochemistry. Relevant sub-discipl... | Geobiology | 0.822333 |
7,776 | Perhaps the oldest of the bunch, paleontology is the study of fossils. It involves the discovery, excavation, dating, and paleoecological understanding of any type of fossil, microbial or dinosaur, trace or body fossil. Micropaleontology is particularly relevant to geobiology. | Geobiology | 0.822333 |
7,777 | These fossilized lipids are called steranes and hopanes, respectively. There are also other types of molecular fossils, like porphyrins, the discovery of which in petroleum by Alfred E. Treibs actually led to the invention of the field. Other aspects of geochemistry that are also pertinent to geobiology include isotope... | Geobiology | 0.822333 |
7,778 | Organic geochemistry is the study of organic molecules that appear in the fossil record in sedimentary rocks. Research in this field concerns molecular fossils that are often lipid biomarkers. Molecules like sterols and hopanoids, membrane lipids found in eukaryotes and bacteria, respectively, can be preserved in the r... | Geobiology | 0.822333 |
7,779 | Molecular geomicrobiology takes a mechanistic approach to understanding biological processes that are geologically relevant. It can be at the level of DNA, protein, lipids, or any metabolite. One example of Molecular geomicrobiology research is studying how recently created lava fields are colonized by microbes. The Un... | Geobiology | 0.822333 |
7,780 | In this way, genes are clues to organismal metabolism and identity. Genetics enables us to ask 'who is there?' and 'what are they doing?' This approach is called metagenomics. | Geobiology | 0.822333 |
7,781 | Molecular biology allows scientists to understand a gene's function using microbial culturing and mutagenesis. Searching for similar genes in other organisms and in metagenomic and metatranscriptomic data allows us to understand what processes could be relevant and important in a given ecosystem, providing insight into... | Geobiology | 0.822333 |
7,782 | Microbiology is a broad scientific discipline pertaining to the study of that life which is best viewed under a microscope. It encompasses several fields that are of direct relevance to geobiology, and the tools of microbiology all pertain to geobiology. Environmental microbiology is especially entangled in geobiology ... | Geobiology | 0.822333 |
7,783 | Many organisms can perform multiple metabolisms to achieve the same end goal; these are called mixotrophs.Biotic metabolism is directly tied to the global cycling of elements and compounds on Earth. The geochemical environment fuels life, which then produces different molecules that go into the external environment. (T... | Geobiology | 0.822333 |
7,784 | Geobiology is founded upon a few core concepts that unite the study of Earth and life. While there are many aspects of studying past and present interactions between life and Earth that are unclear, several important ideas and concepts provide a basis of knowledge in geobiology that serve as a platform for posing resea... | Geobiology | 0.822333 |
7,785 | It deals with process - how organisms affect the Earth and vice versa - as well as history - how the Earth and life have changed together. Much research is grounded in the search for fundamental understanding, but geobiology can also be applied, as in the case of microbes that clean up oil spills.Geobiology employs mol... | Geobiology | 0.822333 |
7,786 | Geobiologic studies tend to be focused on microorganisms, and on the role that life plays in altering the chemical and physical environment of the pedosphere, which exists at the intersection of the lithosphere, atmosphere, hydrosphere and/or cryosphere. It differs from biogeochemistry in that the focus is on processes... | Geobiology | 0.822333 |
7,787 | Geobiology is a field of scientific research that explores the interactions between the physical Earth and the biosphere. It is a relatively young field, and its borders are fluid. There is considerable overlap with the fields of ecology, evolutionary biology, microbiology, paleontology, and particularly soil science a... | Geobiology | 0.822333 |
7,788 | Geobiochemistry is similar to biogeochemistry, but differs by placing emphasis on the effects of geology on the development of life's biochemical processes, as distinct from the role of life on Earth's cycles. Its primary goal is to link biological changes, encompassing evolutionary modifications of genes and changes i... | Geobiology | 0.822333 |
7,789 | This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this ... | Orthogonality relations | 0.822332 |
7,790 | In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: x 2 ≡ q ( mod n ) . {\displaystyle x^{2}\equiv q{\pmod {n}}.} Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical c... | Quadratic residues | 0.822329 |
7,791 | {\sqrt {q}}}}\sum _{n=1}^{\frac {q-1}{2}}\left({\frac {n}{q}}\right)>0.} This implies that there are more quadratic residues than nonresidues among the numbers 1, 2, ..., (q − 1)/2. For example, modulo 11 there are four residues less than 6 (namely 1, 3, 4, and 5), but only one nonresidue (2). An intriguing fact about ... | Quadratic residues | 0.822329 |
7,792 | In essence, the most general form of a ring, is the endomorphism group of some abelian X-group. Any ring can be seen as a preadditive category with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally g... | Mathematical ring | 0.822317 |
7,793 | An epimorphism (that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map Z → Q {\displaystyle \mathbb {Z} \to \mathbb {Q} } is an epimorphism. An algebra homomorphism from a k-algebra to the endomorphism algebra of a vector space over k is called a representation of the algebra.G... | Mathematical ring | 0.822317 |
7,794 | For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal P of R is called a prime ideal if for any elements x , y ∈ R {\displaystyle x,y\in R} we have that x y ∈ P {\displaystyle xy\in P} implies either x ∈ P {\displa... | Mathematical ring | 0.822317 |
7,795 | A ring in which there is no strictly decreasing infinite chain of left ideals is called a left Artinian ring. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the Hopkins–Levitzki theorem). The integers, however, form a Noetherian ring which is not Artinian. | Mathematical ring | 0.822317 |
7,796 | In mathematics, a Stone algebra, or Stone lattice, is a pseudo-complemented distributive lattice such that a* ∨ a** = 1. They were introduced by Grätzer & Schmidt (1957) and named after Marshall Harvey Stone. Boolean algebras are Stone algebras, and Stone algebras are Ockham algebras. Examples: The open-set lattice of ... | Stone algebra | 0.822308 |
7,797 | Maxwell postulated that such waves make up visible light, which was later shown to be true. Indeed, gamma-rays, x-rays, ultraviolet, visible, infrared radiation, microwaves and radio waves were all determined to be electromagnetic radiation differing only in their range of frequencies. In the modern era, scientists hav... | Applied electromagnetics | 0.822307 |
7,798 | In the 18th and 19th centuries, prominent scientists and mathematicians such as Coulomb, Gauss and Faraday developed namesake laws which helped to explain the formation and interaction of electromagnetic fields. This process culminated in the 1860s with the discovery of Maxwell's equations, a set of four partial differ... | Applied electromagnetics | 0.822307 |
7,799 | Meanwhile, magnetic interactions between the spin and angular momentum magnetic moments of electrons also play a role in chemical reactivity; such relationships are studied in spin chemistry. Electromagnetism also plays a crucial role in modern technology: electrical energy production, transformation and distribution; ... | Applied electromagnetics | 0.822307 |
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