When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to vector flux radiance because the macroscopic cross-section @math depends on the distance @math to the previous collision or boundary. This creates a nonlocal relationship between collision rate and flux and requires revising a number of familiar deterministic and Monte Carlo methods. We generalize collision and track-length estimators to support unbiased estimation of either flux integrals or collision rates in generalized radiative transfer (GRT). To provide benchmark solutions for the Monte Carlo estimators, we derive the four Green's functions for the isotropic point source in infinite media with isotropic scattering. Additionally, new moment-preserving diffusion approximations for these Green's functions are derived and reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.
GRT is an efficient and approximate form of scalar stochastic radiative transfer formed by GRT is agnostic to how @math is determined. It may be estimated using Monte Carlo sampling within specific realizations of quenched disorder @cite_1 @cite_32 @cite_39 , by deterministic analysis of a periodic structure @cite_33 , or from observed statistics of the motion of organic material @cite_44 @cite_128 . Alternatively, @math follows from specifying any one of @math , by relationships summarized in Table . The mean attenuation law from a deterministic origin @math may be measured in the laboratory, or calculated from a given stationary random function for @math @cite_50 @cite_142 @cite_6 @cite_14 @cite_106 @cite_69 @cite_70 @cite_13 .
@cite_69 @cite_14 @cite_128 @cite_33 @cite_70 @cite_1 @cite_32 @cite_6 @cite_39 @cite_44 @cite_106 @cite_50 @cite_13 @cite_142
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Abstract Consider a set of n stochastic lines in R 2 , where the existence probability of each line is determined by a fixed probability distribution. For a fixed x -coordinate q , the n lines from top to bottom can be represented by an ordered n -element sequence. Consider all the ( n k ) k -element sub-sequences of that n -element sequence. Each k -element sub-sequence has an associated likelihood to be the true k -topmost lines at x -coordinate q , and the one with the largest probability is defined as the most likely k-topmost lines at q . This paper studies the most likely k -topmost lines of the arrangement of n lines taken over all the x -coordinates. Let cnt be the total number of distinct sequences of the most likely k -topmost lines over all x -coordinates. The main result established is that the expected value of cnt is O ( k n ) , which implies that it is possible to store all the distinct most likely k -topmost lines in O ( k 2 n ) expected space. An example is given showing that cnt , in the worst case, can be Θ ( n 2 ) even when k = 1 . This highlights the value of the expected bound. An algorithm is also given to compute the most likely k -topmost lines of the arrangement. Applications of this result to the stochastic Voronoi Diagram in R 1 and to the stochastic preference top- k query in R 2 are discussed. We consider a discrete elliptic equation with random coefficients @math , which (to fix ideas) are identically distributed and independent from grid point to grid point @math . On scales large w. r. t. the grid size (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. These symmetric ''homogenized'' coefficients @math are characterized by @math where the random field @math is the unique stationary solution of the ''corrector problem'' @math and @math denotes the ensemble average. It is known (''by ergodicity'') that the above ensemble average of the energy density @math , which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of @math on length scales @math is estimated as follows: @math where the averaging function (i. e. @math , @math ) has to be smooth in the sense that @math . In two space dimensions (i. e. @math ), there is a logarithmic correction. In other words, smooth averages of the energy density @math behave like as if @math would be independent from grid point to grid point (which it is not for @math ). This result is of practical significance, since it allows to estimate the error when numerically computing @math . A detailed study is presented of some statistical properties of the stochastic process, that consists of the sum of two sine waves of unknown relative phase and a normal process. Since none of the statistics investigated seem to yield a closed-form expression, all the derivations are cast in a form that is particularly suitable for machine computation. Specifically, results are presented for the probability density function (pdf) of the envelope and the instantaneous value, the moments of these distributions, and the relative cumulative density function (cdf). The analysis hinges on expanding the functions of interest in a way that allows computation by means of recursive relations. Specifically, all the expansions are expressed in terms of sums of products of Gaussian hypergeometric functions and Laguerre polynomials. Computer results obtained on a CDC 6600 are presented. If a and b are the amplitudes of the two sine waves, normalized to the rms noise level, the expansions presented are useful up to values of a,b of about 17 dB, in double precision on the CDC 6600. A different approximation is also given for the case of very high SNR. Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090.