LU decompositionIn numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.
QR decompositionIn linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm. Any real square matrix A may be decomposed as where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning ) and R is an upper triangular matrix (also called right triangular matrix).
Cholesky decompositionIn linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ʃəˈlɛski ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices, and posthumously published in 1924. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.
Singular value decompositionIn linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an complex matrix M is a factorization of the form where U is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an complex unitary matrix, and is the conjugate transpose of V.
Energy transitionAn energy transition (or energy system transformation) is a significant structural change in an energy system regarding supply and consumption. Currently, a transition to sustainable energy (mostly renewable energy) is underway to limit climate change. It is also called renewable energy transition. The current transition is driven by a recognition that global greenhouse-gas emissions must be drastically reduced. This process involves phasing-down fossil fuels and re-developing whole systems to operate on low carbon electricity.
Fossil fuel subsidiesFossil fuel subsidies are energy subsidies on fossil fuels. They may be tax breaks on consumption, such as a lower sales tax on natural gas for residential heating; or subsidies on production, such as tax breaks on exploration for oil. Or they may be free or cheap negative externalities; such as air pollution or climate change due to burning gasoline, diesel and jet fuel. Some fossil fuel subsidies are via electricity generation, such as subsidies for coal-fired power stations.
Fossil fuel phase-outFossil fuel phase-out is the gradual reduction of the use and production of fossil fuels to zero, to reduce deaths and illness from air pollution, limit climate change, and strengthen energy independence. It is part of the ongoing renewable energy transition. Although many countries are shutting down coal-fired power stations, electricity generation is not moving off coal fast enough to meet climate goals. Many countries have set dates to stop selling petrol and diesel cars and trucks, but a timetable to stop burning fossil gas has not yet been agreed.
Fossil fuels lobbyThe fossil fuels lobby includes paid representatives of corporations involved in the fossil fuel industry (oil, gas, coal), as well as related industries like chemicals, plastics, aviation and other transportation. Because of their wealth and the importance of energy, transport and chemical industries to local, national and international economies, these lobbies have the capacity and money to attempt to have outsized influence governmental policy.
Carbon taxA carbon tax is a tax levied on the carbon emissions required to produce goods and services. Carbon taxes are intended to make visible the "hidden" social costs of carbon emissions, which are otherwise felt only in indirect ways like more severe weather events. In this way, they are designed to reduce greenhouse gas emissions by increasing prices of the fossil fuels that emit them when burned. This both decreases demand for goods and services that produce high emissions and incentivizes making them less carbon-intensive.
Fossil fuelA fossil fuel is a hydrocarbon-containing material such as coal, oil, and natural gas, formed naturally in the Earth's crust from the remains of dead plants and animals that is extracted and burned as a fuel. Fossil fuels may be burned to provide heat for use directly (such as for cooking or heating), to power engines (such as internal combustion engines in motor vehicles), or to generate electricity. Some fossil fuels are refined into derivatives such as kerosene, gasoline and propane before burning.
Fossil fuel power stationA fossil fuel power station is a thermal power station which burns a fossil fuel, such as coal or natural gas, to produce electricity. Fossil fuel power stations have machinery to convert the heat energy of combustion into mechanical energy, which then operates an electrical generator. The prime mover may be a steam turbine, a gas turbine or, in small plants, a reciprocating gas engine. All plants use the energy extracted from the expansion of a hot gas, either steam or combustion gases.
Polar decompositionIn mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form , where is a unitary matrix and is a positive semi-definite Hermitian matrix ( is an orthogonal matrix and is a positive semi-definite symmetric matrix in the real case), both square and of the same size. Intuitively, if a real matrix is interpreted as a linear transformation of -dimensional space , the polar decomposition separates it into a rotation or reflection of , and a scaling of the space along a set of orthogonal axes.
Efficient energy useEfficient energy use, sometimes simply called energy efficiency, is the process of reducing the amount of energy required to provide products and services. For example, insulating a building allows it to use less heating and cooling energy to achieve and maintain a thermal comfort. Installing light-emitting diode bulbs, fluorescent lighting, or natural skylight windows reduces the amount of energy required to attain the same level of illumination compared to using traditional incandescent light bulbs.
Petroleum productPetroleum products are materials derived from crude oil (petroleum) as it is processed in oil refineries. Unlike petrochemicals, which are a collection of well-defined usually pure organic compounds, petroleum products are complex mixtures. Most petroleum is converted into petroleum products, which include several classes of fuels. According to the composition of the crude oil and depending on the demands of the market, refineries can produce different shares of petroleum products.
Fossil fuel divestmentFossil fuel divestment or fossil fuel divestment and investment in climate solutions is an attempt to reduce climate change by exerting social, political, and economic pressure for the institutional divestment of assets including stocks, bonds, and other financial instruments connected to companies involved in extracting fossil fuels. Fossil fuel divestment campaigns emerged on campuses in the United States in 2011 with students urging their administrations to turn endowment investments in the fossil fuel industry into investments in clean energy and communities most impacted by climate change.
Carbon priceCarbon pricing (or pricing) is a method for nations to address climate change. The cost is applied to greenhouse gas emissions in order to encourage polluters to reduce the combustion of coal, oil and gas – the main driver of climate change. The method is widely agreed and considered to be efficient. Carbon pricing seeks to address the economic problem that emissions of and other greenhouse gases (GHG) are a negative externality – a detrimental product that is not charged for by any market.
PetroleumPetroleum, also known as crude oil, or simply oil, is a naturally occurring yellowish-black liquid mixture of mainly hydrocarbons, and is found in geological formations. The name petroleum covers both naturally occurring unprocessed crude oil and petroleum products that consist of refined crude oil. A fossil fuel, petroleum is formed when large quantities of dead organisms, mostly zooplankton and algae, are buried underneath sedimentary rock and subjected to both prolonged heat and pressure.
Transition metalIn chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. The lanthanide and actinide elements (the f-block) are called inner transition metals and are sometimes considered to be transition metals as well. Since they are metals, they are lustrous and have good electrical and thermal conductivity.
Phase transitionIn chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter: solid, liquid, and gas, and in rare cases, plasma. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure.
Energy subsidyEnergy subsidies are measures that keep prices for customers below market levels, or for suppliers above market levels, or reduce costs for customers and suppliers. Energy subsidies may be direct cash transfers to suppliers, customers, or related bodies, as well as indirect support mechanisms, such as tax exemptions and rebates, price controls, trade restrictions, and limits on market access. The International Renewable Energy Agency tracked some $634 billion in energy-sector subsidies in 2020, and found that around 70% were fossil fuel subsidies.
