• ###### 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)
• ###### 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)
• ###### 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)
• ###### 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)
• ###### uniform
• RHA is called homogeneous armour because its structure and composition is uniform throughout its thickness. (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)
• ###### 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)