We study combinatorics of billiard partitions which arose recently in the description of periodic trajectories of ellipsoidal billiards in
d
-dimensional Euclidean and pseudo-Euclidean spaces. Such ...partitions uniquely codify the sets of caustics, up to their types, which generate periodic trajectories. The period of a periodic trajectory is the largest part while the winding numbers are the remaining summands of the corresponding partition. In order to take into account the types of caustics as well, we introduce weighted partitions and provide closed forms for the generating functions of these partitions.
We study four-derivative corrections to four-dimensional N = 2 minimal gauged supergravity controlled by two real constants. The solutions of the equations of motion in the two-derivative theory are ...not modified by the higher-derivative corrections. We use this to derive a general formula for the regularized on-shell action for any asymptotically locally AdS4 solution of the theory and show how the higher-derivative corrections affect black hole thermodynamic quantities in a universal way. We employ our results in the context of holography to derive explicit expressions for the subleading corrections in the large N expansion of supersymmetric partition functions on various compact manifolds for a large class of three-dimensional SCFTs.
Random Partition Models for Microclustering Tasks Betancourt, Brenda; Zanella, Giacomo; Steorts, Rebecca C.
Journal of the American Statistical Association,
09/2022, Letnik:
117, Številka:
539
Journal Article
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Traditional Bayesian random partition models assume that the size of each cluster grows linearly with the number of data points. While this is appealing for some applications, this assumption is not ...appropriate for other tasks such as entity resolution (ER), modeling of sparse networks, and DNA sequencing tasks. Such applications require models that yield clusters whose sizes grow sublinearly with the total number of data points-the microclustering property. Motivated by these issues, we propose a general class of random partition models that satisfy the microclustering property with well-characterized theoretical properties. Our proposed models overcome major limitations in the existing literature on microclustering models, namely a lack of interpretability, identifiability, and full characterization of model asymptotic properties. Crucially, we drop the classical assumption of having an exchangeable sequence of data points, and instead assume an exchangeable sequence of clusters. In addition, our framework provides flexibility in terms of the prior distribution of cluster sizes, computational tractability, and applicability to a large number of microclustering tasks. We establish theoretical properties of the resulting class of priors, where we characterize the asymptotic behavior of the number of clusters and of the proportion of clusters of a given size. Our framework allows a simple and efficient Markov chain Monte Carlo algorithm to perform statistical inference. We illustrate our proposed methodology on the microclustering task of ER, where we provide a simulation study and real experiments on survey panel data.
In this note, we consider the number of k’s in all the partitions of n in order to provide a new proof of a classical identity involving Euler’s partition function p(n) and the sum of the positive ...divisors function a(n). New relations connecting classical functions of multiplicative number theory with the partition function p(n) from additive number theory are introduced in this context. The fascinating feature of these relations is their common nature. A new identity for the number of 1’s in all the partitions of n is derived in this context.
In traditional steganographic schemes, RGB three channels payloads are assigned equally in a true color image. In fact, the security of color image steganography relates not only to data-embedding ...algorithms but also to different payload partition. How to exploit inter-channel correlations to allocate payload for performance enhancement is still an open issue in color image steganography. In this paper, a novel channel-dependent payload partition strategy based on amplifying channel modification probabilities is proposed, so as to adaptively assign the embedding capacity among RGB channels. The modification probabilities of three corresponding pixels in RGB channels are simultaneously increased, and thus the embedding impacts could be clustered, in order to improve the empirical steganographic security against the channel co-occurrences detection. The experimental results show that the new color image steganographic schemes, incorporated with the proposed strategy, can effectively make the embedding changes concentrated mainly in textured regions, and achieve better performance on resisting the modern color image steganalysis.
Levels of polybrominated diphenyl ethers (PBDEs) and dechlorane plus (DPs) were investigated in the Indus River Basin from Pakistan. Concentrations of ∑PBDEs and ∑DPs were ranged between 0.05 and ...2.38 and 0.002–0.53 ng g−1 in the surface soils while 1.43–22.1 and 0.19–7.59 pg m−3 in the passive air samples, respectively. Black carbon (fBC) and total organic carbon (fTOC) fractions were also measured and ranged between 0.73 and 1.75 and 0.04–0.2%, respectively. The statistical analysis revealed strong influence of fBC than fTOC on the distribution of PBDEs and DPs in the Indus River Basin soils. BDE's congener profile suggested the input of penta–bromodiphenylether (DE-71) commercial formulation in the study area. Soil–air partitioning of PBDEs were investigated by employing octanol-air partition coefficients (KOA) and black carbon-air partition coefficients (KBC−A). The results of both models suggested the combined influence of total organic carbon (absorption) and black carbon (adsorption) in the studied area.
•Model based calculations of black carbon-air partition coefficients for PBDEs.•Soil and air levels of PBDEs and DPs reported first time for ecologically important sites of the Indus River Basin, Pakistan.•Both, fBC and fTOC showed combined influence on soil–air partitioning of PBDEs in the Indus River Basin, Pakistan.
BC and TOC showed combined influence on soil–air partitioning of POPs i-e., PBDEs in the Indus River Basin, Pakistan.
This work focuses on fuzzy orthopartitions and credal partitions, which are distinct mathematical models representing partitions where the membership of elements to classes is only partially known. ...Firstly, we show that fuzzy orthopartitions and credal partitions are special cases of generalized fuzzy orthopartitions, which we introduce in this article as a new structure for modelling partitions with uncertainty. Next, we examine the connections between credal partitions and fuzzy orthopartitions, considering that both can be seen as types of fuzzy partitions (in particular, we deal with fuzzy probabilistic and Ruspini partitions). Moreover, we find that each generalized fuzzy orthopartition corresponds to a collection of zero, one, or infinitely many credal partitions; conversely, a credal partition maps to at most one generalized fuzzy orthopartition. Finally, we identify the class of all credal partitions that coincide with fuzzy orthopartitions.
This paper provides some new and novel application-independent perspectives on why improved performance usually occurs as one goes from crisp, to type-1 (T1), and to interval type-2 (IT2) fuzzy ...systems, by introducing three kinds of partitions: (1) Uncertainty partitions that let us distinguish T1 fuzzy sets from crisp sets, and IT2 fuzzy sets from T1 fuzzy sets; (2) First-and second-order rule partitions that are direct results of uncertainty partitions, and are associated with the number of rules that fire in different regions of the state space, and, the changes in their mathematical formulae within those regions; and (3) Novelty partitions that can only occur in an IT2 fuzzy system that uses type-reduction. Rule and novelty partitions sculpt the state space into hyperrectangles within each of which resides a different nonlinear function. It is the author's conjecture that the greater sculpting of the state space by a T1 fuzzy system lets it outperform a crisp system, and the even greater sculpting of the state space by an IT2 fuzzy system lets it outperform a T1 fuzzy system. The latter can occur even when the T1 and IT2 fuzzy systems are described by the same number of parameters.
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