###### catalysis

- The Homogeneous Catalysis Award recognises outstanding research in homogeneous catalysis. (rsc.org)
- The RSC Homogeneous Catalysis Award was established in 2001. (rsc.org)
- We are relating supramolecular and molecular recognition chemistry to homogeneous catalysis. (bio.net)
- Boy Cornils has worked at the Ruhrchemie AG in Oberhausen, Germany, where homogeneous catalysis started with the hydroformylation reaction found by Otto Roelen, and with the former Hoechst AG in Frankfurt, Germany. (wiley-vch.de)
- To address the relationships between homogeneous and heterogeneous catalysis in a creative environment, where all can learn from and be inspired by each other's dreams, successes and breakthroughs. (rsc.org)
- In chemistry, homogeneous catalysis is catalysis in a solution by a soluble catalyst. (wikipedia.org)
- Strictly speaking, homogeneous catalysis refers to catalytic reactions where the catalyst is in the same phase as the reactants. (wikipedia.org)
- Homogeneous catalysis applies to reactions in the gas phase and even in solids. (wikipedia.org)
- Heterogeneous catalysis is the alternative to homogeneous catalysis, where the catalysis occurs at the interface of two phases, typically gas-solid. (wikipedia.org)
- Processes that utilize soluble organometallic compounds as catalysts fall under the category of homogeneous catalysis, as opposed to processes that use bulk metal or metal on a solid support, which are examples of heterogeneous catalysis. (wikipedia.org)
- Some well-known examples of homogeneous catalysis include hydroformylation and transfer hydrogenation, as well as certain kinds of Ziegler-Natta polymerization and hydrogenation. (wikipedia.org)
- Homogeneous catalysis differs from heterogeneous catalysis in that the catalyst is in a different phase than the reactants. (wikipedia.org)
- Heterogeneous catalysis offers the advantage that products are readily separated from the catalyst, and heterogeneous catalysts are often more stable and degrade much slower than homogeneous catalysts. (wikipedia.org)

###### polynomial of degree

- The dimension of the vector space (or free module) R d {\displaystyle R_{d}} is the number of different monomials of degree d in n variables (that is the maximal number of nonzero terms in a homogeneous polynomial of degree d in n variables). (wikipedia.org)
- Given a homogeneous polynomial of degree k, it is possible to get a homogeneous function of degree 1 by raising to the power 1/k. (wikipedia.org)

###### mathematics

- In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. (wikipedia.org)
- In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry. (wikipedia.org)
- In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree. (wikipedia.org)
- Homogeneous polynomials are ubiquitous in mathematics and physics. (wikipedia.org)
- In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor. (wikipedia.org)
- In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. (wikipedia.org)

###### Euler's

- Furthermore, a version of Euler's homogeneous function theorem holds: a distribution S is homogeneous of degree α if and only if ∑ i = 1 n x i ∂ S ∂ x i = α S . {\displaystyle \sum _{i=1}^{n}x_{i}{\frac {\partial S}{\partial x_{i}}}=\alpha S.} A complete classification of homogeneous distributions in one dimension is possible. (wikipedia.org)
- Homogeneous polynomial satisfy Euler's identity for homogeneous functions. (wikipedia.org)
- 0. Positively homogeneous functions are characterized by Euler's homogeneous function theorem. (wikipedia.org)

###### displaystyle

- This shows us why we can construct S n − 1 {\displaystyle S^{n-1}} as a homogeneous space. (wikipedia.org)
- T {\displaystyle T} is said to be κ {\displaystyle \kappa } -homogeneous if each μ s {\displaystyle \mu _{s}} is κ {\displaystyle \kappa } -complete. (wikipedia.org)
- If S is a homogeneous distribution on Rn \ {0} of degree α, then the weak first partial derivative of S ∂ S ∂ x i {\displaystyle {\frac {\partial S}{\partial x_{i}}}} has degree α−1. (wikipedia.org)
- Throughout the domain of definition, xα + also satisfies the following properties: d d x x + α = α x + α − 1 {\displaystyle {\frac {d}{dx}}x_{+}^{\alpha }=\alpha x_{+}^{\alpha -1}} x x + α = x + α + 1 {\displaystyle xx_{+}^{\alpha }=x_{+}^{\alpha +1}} There are several distinct ways to extend the definition of power functions to homogeneous distributions on R at the negative integers. (wikipedia.org)
- The polynomial x 3 + 3 x 2 y + z 7 {\displaystyle x^{3}+3x^{2}y+z^{7}} is not homogeneous, because the sum of exponents does not match from term to term. (wikipedia.org)
- The natural logarithm f ( x ) = ln x {\displaystyle f(x)=\ln x} scales additively and so is not homogeneous. (wikipedia.org)
- One way to follow basis-dependence in a linear algebra argument is to track variables x in X. Similarly, the space of orthonormal bases (the Stiefel manifold V n ( R n ) {\displaystyle V_{n}(\mathbf {R} ^{n})} of n-frames) is a principal homogeneous space for the orthogonal group. (wikipedia.org)
- The term quasi-homogeneous comes from the fact that a polynomial f is quasi-homogeneous if and only if f ( λ w 1 x 1 , … , λ w r x r ) = λ w f ( x 1 , … , x r ) {\displaystyle f(\lambda ^{w_{1}}x_{1},\ldots ,\lambda ^{w_{r}}x_{r})=\lambda ^{w}f(x_{1},\ldots ,x_{r})} for every λ {\displaystyle \lambda } in any field containing the coefficients. (wikipedia.org)
- In other words, a polynomial is quasi-homogeneous if all the α {\displaystyle \alpha } belong to the same affine hyperplane. (wikipedia.org)
- A first-order ordinary differential equation in the form: M ( x , y ) d x + N ( x , y ) d y = 0 {\displaystyle M(x,y)\,dx+N(x,y)\,dy=0} is a homogeneous type if both functions M(x, y) and N(x, y) are homogeneous functions of the same degree n. (wikipedia.org)
- A linear differential equation is called homogeneous if the following condition is satisfied: If ϕ ( x ) {\displaystyle \phi (x)} is a solution, so is c ϕ ( x ) {\displaystyle c\phi (x)} , where c {\displaystyle c} is an arbitrary (non-zero) constant. (wikipedia.org)

###### Symmetric

- The structure of a G-space is a group homomorphism ρ : G → Diffeo(X) into the diffeomorphism group of X. Riemannian symmetric spaces are an important class of homogeneous spaces, and include many of the examples listed below. (wikipedia.org)
- Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. (wikipedia.org)
- In finite dimensions, they establish an isomorphism of graded vector spaces from the symmetric algebra of V∗ to the algebra of homogeneous polynomials on V. Rational functions formed as the ratio of two homogeneous polynomials are homogeneous functions off of the affine cone cut out by the zero locus of the denominator. (wikipedia.org)

###### heterogeneous

- Enzymes possess properties of both homogeneous and heterogeneous catalysts. (wikipedia.org)
- Homogeneous" and "heterogeneous" are not absolute terms, but depend on context and size of the sample. (wikipedia.org)

###### Catalysts

- Enzymes are examples of homogeneous catalysts. (wikipedia.org)
- Homogeneous catalysts have also been used in a variety of industrial processes, such as the Wacker process Acetaldehyde (conversion of ethylene to acetaldehyde) as well as the Monsanto process and the Cativa process for the conversion of MeOH and CO to acetic acid. (wikipedia.org)
- Enzymes are homogeneous catalysts that are essential for life but are also harnessed for industrial processes. (wikipedia.org)
- http://goldbook.iupac.org/C00876.html P. W. N. M. van Leeuwen and J. C. Chadwick "Homogeneous Catalysts: Activity - Stability - Deactivation" Wiley-VCH, Weinheim, 2011. (wikipedia.org)

###### coordinates

- Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. (wikipedia.org)
- If the homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. (wikipedia.org)
- Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than the dimension of the projective space being considered. (wikipedia.org)
- For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point in the projective plane. (wikipedia.org)
- Given a point (x, y) on the Euclidean plane, for any non-zero real number Z, the triple (xZ, yZ, Z) is called a set of homogeneous coordinates for the point. (wikipedia.org)
- By this definition, multiplying the three homogeneous coordinates by a common, non-zero factor gives a new set of homogeneous coordinates for the same point. (wikipedia.org)
- Thus unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates. (wikipedia.org)
- Thus we define (m, −n, 0) as the homogeneous coordinates of the point at infinity corresponding to the direction of the line nx + my = 0. (wikipedia.org)
- As any line of the Euclidean plane is parallel to a line passing through the origin, and since parallel lines have the same point at infinity, the infinite point on every line of the Euclidean plane has been given homogeneous coordinates. (wikipedia.org)
- To summarize: Any point in the projective plane is represented by a triple (X, Y, Z), called the homogeneous coordinates or projective coordinates of the point, where X, Y and Z are not all 0. (wikipedia.org)
- The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor. (wikipedia.org)
- Conversely, two sets of homogeneous coordinates represent the same point if and only if one is obtained from the other by multiplying all the coordinates by the same non-zero constant. (wikipedia.org)
- Some authors use different notations for homogeneous coordinates which help distinguish them from Cartesian coordinates. (wikipedia.org)
- The use of real numbers gives the homogeneous coordinates of points in the classical case of the real projective spaces, however any field may be used, in particular, the complex numbers may be used for complex projective space. (wikipedia.org)
- A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. (wikipedia.org)
- The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space). (wikipedia.org)
- The correspondence between homogeneous ideals I and varieties is bijective for ideals not containing the ideal J generated by all the Xi, which corresponds to the empty set because not all homogeneous coordinates can vanish at a point of projective space. (wikipedia.org)

###### projective space

- Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for their respective symmetry groups. (wikipedia.org)
- 0. Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of sheaves on projective space in algebraic geometry. (wikipedia.org)

###### finite

- The only finite homogeneous graphs are the cluster graphs mKn formed from the disjoint unions of isomorphic complete graphs, the Turán graphs formed as the complement graphs of mKn, the 3 × 3 rook's graph, and the 5-cycle. (wikipedia.org)
- In addition to the homogeneous graphs, the finite connected-homogeneous graphs include all cycle graphs, all square rook's graphs, the Petersen graph, and the 5-regular Clebsch graph. (wikipedia.org)

###### algebraic variety

- In algebraic geometry, a homogeneous variety is an algebraic variety of the form G/P, G a linear algebraic group, P a parabolic subgroup. (wikipedia.org)
- They play a fundamental role in algebraic geometry, as a projective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials. (wikipedia.org)

###### defines

- A polynomial is homogeneous if and only if it defines a homogeneous function. (wikipedia.org)
- A homogeneous polynomial defines a homogeneous function. (wikipedia.org)

###### nonzero

- Any nonzero polynomial may be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees, which are called the homogeneous components of the polynomial. (wikipedia.org)

###### intuitively

- In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). (wikipedia.org)

###### Definition

- The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero nilpotent elements and other divisors of zero. (wikipedia.org)
- This property of being intrinsic to R allows the definition of the graded Betti numbers, namely the βi, j which are the number of grade-j images coming from Fi (more precisely, by thinking of φ as a matrix of homogeneous polynomials, the count of entries of that homogeneous degree incremented by the gradings acquired inductively from the right). (wikipedia.org)

###### mixture

- A homogeneous substance can be a mixture, a compound or an element. (reference.com)
- An example of a homogeneous mixture is salt water in which the salt has completely dissolved. (reference.com)
- A homogeneous mixture is a solid, liquid or gaseous mixture that has the same proportions of its components throughout a given sample (or multiple samples of different proportion). (wikipedia.org)
- An example of a homogeneous mixture is air. (wikipedia.org)
- A solution is a special type of homogeneous mixture. (wikipedia.org)
- As a homogeneous mixture, a solution has one phase (liquid) although the solute and solvent can vary: for example, salt water. (wikipedia.org)
- Homogeneous mixtures have the same proportions of the various components throughout a given sample (or multiple samples of different proportion), creating a consistent mixture. (wikipedia.org)
- In an HCCI engine, however, the homogeneous mixture of fuel and air is compressed and combustion begins whenever sufficient pressure and temperature are reached. (wikipedia.org)

###### Thus

- Thus a homogeneous space can be thought of as a coset space without a choice of origin. (wikipedia.org)
- Thus, if f is homogeneous of degree m and g is homogeneous of degree n, then f/g is homogeneous of degree m − n away from the zeros of g. (wikipedia.org)
- The fuel is mixed with the coolant and the moderator, thus the name "homogeneous" ('of the same physical state') The water can be either heavy water or ordinary (light) water, both of which need to be very pure. (wikipedia.org)

###### spaces

- Higher order contact of submanifolds on homogeneous spaces , Lectures notes in Math. (springer.com)
- Szenthe, J.: Transformations groups on homogeneous spaces, Rend. (springer.com)
- Villarroel Y. (1996) Totally geodesic orbits in homogeneous spaces. (springer.com)
- Blok R.J., Cardinali I., De Bruyn B., Pasini A.: Polarized and homogeneous embeddings of dual polar spaces. (springer.com)
- It follows that the n-th differential of a function ƒ : X → Y between two Banach spaces X and Y is homogeneous of degree n. (wikipedia.org)

###### chemistry

- In chemistry, a homogeneous suspension of material means that when dividing the volume in half, the same amount of material is suspended in both halves of the substance. (wikipedia.org)

###### equation

- A differential equation can be homogeneous in either of two respects: the coefficients of the differential terms in the first order case could be homogeneous functions of the variables, or for the linear case of any order there could be no constant term. (wikipedia.org)

###### Conversely

- Conversely, given a coset space G/H, it is a homogeneous space for G with a distinguished point, namely the coset of the identity. (wikipedia.org)

###### given

- It can be a non-trivial problem to extend a given homogeneous distribution from Rn \ {0} to a distribution on Rn, although this is necessary for many of the techniques of Fourier analysis, in particular the Fourier transform, to be brought to bear. (wikipedia.org)
- The homogeneous distributions on R \ {0} are given by various power functions. (wikipedia.org)
- A k-homogeneous graph obeys a weakened version of the same property in which every isomorphism between two induced subgraphs implies the existence of an automorphism of the whole graph that maps one subgraph to the other (but does not necessarily extend the given isomorphism). (wikipedia.org)

###### ratio

- Solutions are homogeneous because the ratio of solute to solvent remains the same throughout the solution even if homogenized with multiple sources, and stable because the solute will not settle out after any period of time, and it cannot be removed by a filter or by centrifuge. (wikipedia.org)

###### stabilizer

- In general, if X is a homogeneous space, and Ho is the stabilizer of some marked point o in X (a choice of origin), the points of X correspond to the left cosets G/Ho, and the marked point o corresponds to the coset of the identity. (wikipedia.org)

###### type

- We construct a homogeneous full projective embedding of the dual polar space \(\textit{DW}(2n-1,2)\) from the hyperplane intersections of hyperbolic type of the parabolic quadric Q (2 n , 2). (springer.com)
- Homogeneous broadening is a type of emission spectrum broadening in which all atoms radiating from a specific level under consideration radiate with equal opportunity. (wikipedia.org)
- If the fluctuation in the frequency/wavelength is due to a phenomenon that is the same for each quantum emitter, there is homogeneous broadening, while if each quantum emitter has a different type of fluctuation, the broadening is inhomogeneous. (wikipedia.org)
- Aqueous homogeneous reactors (AHR) are a type of nuclear reactor in which soluble nuclear salts (usually uranium sulfate or uranium nitrate) are dissolved in water. (wikipedia.org)
- Rolled homogeneous armour (RHA) is a type of armour made of a single steel composition hot-rolled to improve its material characteristics, as opposed to layered or cemented armour. (wikipedia.org)
- For the testing and calibration of anti-tank guns, the term RHAe (Rolled Homogeneous Armour equivalency) is used when giving an estimate of either the penetrative capability of a projectile or the protective capability of a type of armor which may or may not be steel. (wikipedia.org)

###### degree

- The Dirac delta function is homogeneous of degree −1. (wikipedia.org)
- Moreover, the kth weak derivative of the delta function δ(k) is homogeneous of degree −k−1. (wikipedia.org)
- The distribution is homogeneous of degree α. (wikipedia.org)
- alpha }=x_{+}^{\alpha }+x_{-}^{\alpha }} are homogeneous distributions of degree α. (wikipedia.org)
- A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. (wikipedia.org)

###### Space

- A homogeneous space is a G-space on which G acts transitively. (wikipedia.org)
- The geometry of the resulting homogeneous space is the line geometry of Julius Plücker. (wikipedia.org)
- The composition of this operation with the right group action, however, yields a ternary operation X × (X × X) → X × G → X that serves as an affine generalization of group multiplication and is sufficient to both characterize a principal homogeneous space algebraically, and intrinsically characterize the group it is associated with. (wikipedia.org)
- Another example is the affine space concept: the idea of the affine space A underlying a vector space V can be said succinctly by saying that A is a principal homogeneous space for V acting as the additive group of translations. (wikipedia.org)
- What is more, a linear transformation fixing each vector of a basis will fix all v in V, hence being the neutral element of the general linear group GL(V) : so that X is indeed a principal homogeneous space. (wikipedia.org)
- It is homogeneous which means the statistical characteristics of turbulence is independent of location of space. (wikipedia.org)

###### solvent

- The proton is the most pervasive homogeneous catalyst because water is the most common solvent. (wikipedia.org)

###### However

- However, as some authors do not make a clear distinction between a polynomial and its associated function, the terms homogeneous polynomial and form are sometimes considered as synonymous. (wikipedia.org)
- However, two homogeneous mixtures of the same pair of substances may differ widely from each other and can be homogenized to make a constant. (wikipedia.org)

###### functions

- In addition to the power functions, homogeneous distributions on R include the Dirac delta function and its derivatives. (wikipedia.org)
- Homogeneous polynomials also define homogeneous functions. (wikipedia.org)

###### uniform

- RHA is called homogeneous armour because its structure and composition is uniform throughout its thickness. (wikipedia.org)

###### composition

- Homogeneous mixtures can have a variable composition. (wikipedia.org)
- Homogeneous mixtures always have the same composition. (wikipedia.org)

###### always

- In particular, a homogeneous polynomial is always quasi-homogeneous, with all weights equal to 1. (wikipedia.org)

###### point

- But, they said, the world of food has become homogeneous, to the point of suggesting a global standard food supply. (latimes.com)
- Homogeneous charge compression ignition (HCCI) is a form of internal combustion in which well-mixed fuel and oxidizer (typically air) are compressed to the point of auto-ignition. (wikipedia.org)

###### sometimes

- In liquid crystals homogeneous alignment, sometimes called planar alignment, is the state of alignment where molecules align in parallel to a substrate. (wikipedia.org)
- Aqueous homogeneous reactors were sometimes called "water boilers" (not to be confused with boiling water reactors), as the water inside seems to boil, but in fact this bubbling is due to the production of hydrogen and oxygen as radiation and fission particles dissociate the water into its constituent gases, a process called radiolysis. (wikipedia.org)

###### form

- An algebraic form, or simply form, is a function defined by a homogeneous polynomial. (wikipedia.org)
- These two constructions, one of a homogeneous polynomial from a multilinear form and the other of a multilinear form from a homogeneous polynomial, are mutually inverse to one another. (wikipedia.org)

###### Classical

- Rates were studied in the range 3 × 10−1 − 3 × 102 cm−3 s−1 at four isotherms: 290, 300, 310, and 320 K. Measured isothermal dependencies of nucleation rate of water on supersaturation were compared with prediction of classical theory of homogeneous nucleation. (springer.com)

###### system

- The ImageXpress Velos System provides an ideal platform for a fast, simple, and high-throughput homogeneous (no wash) cytotoxicity assay. (moleculardevices.com)
- Examples of situations where the fluctuation is the same for each system (homogeneous broadening) are natural or lifetime broadening, and collisional or pressure broadening. (wikipedia.org)

###### natural

- If an optical emitter (e.g. an atom) shows homogeneous broadening, its spectral linewidth is its natural linewidth, with a Lorentzian profile. (wikipedia.org)