Problems and Snapshots from the World of Probability

Blom, Gunnar; Holst, Lars; Sandell, Dennis

doi:10.1007/978-1-4612-4304-5

Cited by 87 publications

(47 citation statements)

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“…is the simplest to apply [35]. Domain by domain, one evaluates the difference between the uniform-predicted probability of occurrence of a nanotube and the observed probability.…”

Section: Methodsmentioning

confidence: 99%

Relationship between dispersion metric and properties of PMMA/SWNT nanocomposites

Kashiwagi

Fagan

Douglas

et al. 2007

Polymer

169

113

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Particle spatial dispersion is a crucial characteristic of polymer composite materials and this property is recognized as especially important in nanocomposite materials due to the general tendency of nanoparticles to aggregate under processing conditions. We introduce dispersion metrics along with a specified dispersion scale over which material homogeneity is measured and consider how the dispersion metrics correlate quantitatively with the variation of basic nanocomposite properties. We then address the general problem of quantifying nanoparticle spatial dispersion in model nanocomposites of single wall carbon nanotubes (SWNT) dispersed in poly(methyl methacrylate) (PMMA) at a fixed SWNT concentration of 0.5 % using a 'coagulation' fabrication method. Two methods are utilized to measure dispersion, UV-Vis spectroscopy and optical confocal microscopy. Quantitative spatial dispersion levels were obtained through image analysis to obtain a 'relative dispersion index' (RDI) representing the uniformity of the dispersion of SWNTs in the samples and through absorbance. We find that the storage modulus, electrical conductivity, and flammability containing the same amount of SWNTs, the relationships between the quantified dispersion levels and physical properties show about four orders of magnitude variation in storage modulus, almost eight orders of magnitude variation in electric conductivity, and about 70 % reduction in peak mass loss rate at the highest dispersion level used in this study. The observation of such a profound effect of SWNT dispersion indicates the need for objective dispersion metrics for correlating and understanding how the properties of nanocomposites are determined by the concentration, shape and size of the nanotubes. conditions. We introduce dispersion metrics along with a specified dispersion scale over which material homogeneity is measured and consider how the dispersion metrics correlate quantitatively with the variation of basic nanocomposite properties. We then address the general problem of quantifying nanoparticle spatial dispersion in model nanocomposites of single wall carbon nanotubes (SWNT) dispersed in poly(methyl methacrylate) (PMMA) at a fixed SWNT concentration of 0.5 % using a 'coagulation' fabrication method. Two methods are utilized to measure dispersion, UV-Vis spectroscopy and optical confocal microscopy. Quantitative spatial dispersion levels were obtained through image analysis to obtain a 'relative dispersion index ' (RDI) representing the uniformity of the dispersion of SWNTs in the samples and through absorbance. We find that the storage modulus, electrical conductivity, and flammability † This was carried out by the National Institute of Standards and Technology (NIST), an agency of the US Government and is not subject to copyright in the US. ‡ Correspondence to: T. Kashiwagi (E-mail: takashi.kashiwagi@nist.gov) 1 property of the nanocomposites correlates well with the RDI. For the nanocomposites containing the same amount of SWNTs, the relationships betw...

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“…is the simplest to apply [35]. Domain by domain, one evaluates the difference between the uniform-predicted probability of occurrence of a nanotube and the observed probability.…”

Section: Methodsmentioning

confidence: 99%

Relationship between dispersion metric and properties of PMMA/SWNT nanocomposites

Kashiwagi

Fagan

Douglas

et al. 2007

Polymer

169

113

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“…The expectation of the length of a random sequence covering all w-mers at least once may be approximated by E[L] ≈ 4 w · (ln(4 w ) + 0.577) (Blom et al, 1994), where the factor ln(4 w ) + 0.577 amounts to approximately 8.9 for w = 6 and 14.4 for w = 10.…”

Section: /10mentioning

confidence: 99%

Motif clustering with implications for transcription factor interactions

Grau

Große

Posch

et al. 2015

Preprint

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High-throughput data, for instance ChIP-seq data, measure binding of transcription factors (TFs) or other proteins to DNA and have become a widespread data source for de-novo motif discovery. Often, several ChIP-seq data sets study the same TF under different conditions resulting in several, potentially redundant motifs, which demands for identification and clustering of similar motifs. Here, we propose a refined measure of motif similarity based on the correlation between score profiles on de Bruijn sequences. We demonstrate the utility of the proposed measure in benchmark studies on artificial motifs and motifs discovered from ENCODE ChIP-seq data. We use this measure to cluster motifs discovered from 757 different ENCODE ChIP-seq data sets for 166 TFs and RNA-polymerase II and III. Based on this clustering, we derive a TF interaction network that reflects many known TF-TF interactions, but also reveals novel putative interaction partners.

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“…This situation is stated formally as follows: This corollary holds for a slightly wider condition of sampling as shown below. Let us remind first the fact that the probability of a certain q-gram g appearing in a text does not depend on the position in the text as long as the text is composed of characters randomly chosen from Σ with/without replacement, or equivalently, when we consider all possible permutations of a fixed text [20]. The probability g is found in any position in t, denoted by Pr(g), is the product of matching probabilities of all characters of g. Especially the probability is 1/|Σ| q when all characters appear in the same probability 1/|Σ|.…”

Section: Algorithm Array+base-searchmentioning

confidence: 99%

“…For example, if we want to find the position c i = 15 or the next larger one in Fig. 6, we first search for c i in T by ValueRange (starting from the root (6,10] and then moving to the right and finding (10,20]), then search the chunk A [8..11], specified by IndexRange, with binary searching. Note that the array index does not cause O(k)-time element insertion since element insertions are done only at the end in the algorithm (explained later).…”

Section: Algorithm List+base-search 431 Outlinementioning

confidence: 99%

Average-case linear-time similar substring searching by the q -gram distance

Hanada

Kudo

Nakamura

2014

Theoretical Computer Science

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In this paper we consider the problem of similar substring searching in the q-gram distance. The q-gram distance d q (x, y) is a similarity measure between two strings x and y defined by the number of different q-grams between them. The distance can be used instead of the edit distance due to its lower computation cost, O(|x| + |y|) vs. O(|x||y|), and its good approximation for the edit distance. However, if this distance is applied to the problem of finding all similar strings, in a long text t, to a given pattern p, the total computation cost is sometimes not acceptable. Ukkonen already proposed two fast algorithms: one with an array and the other with a tree. When "similar" means k or less in d q , their time complexities are O(|t|k + |p|) and O(|t| log k +|p|), respectively. In this paper, we propose two algorithms of averagecase complexity O(|t|+|p|), although their worst-case complexities are still O(|t|k+ |p|) and O(|t| log k +|p|), respectively. The linearity of the average-case complexity is analyzed under the assumption of random sampling of t and the condition that q is larger than a threshold. The algorithms exploit the fact that similar substrings in t are often found at very close positions if the beginning positions of the substrings are close. In the second proposed algorithm, we adopted a doubly-linked list supported by an array and a search tree to search for a list element in O(log k) time. Experimental results supported their theoretical average-case complexities.

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Problems and Snapshots from the World of Probability

Cited by 87 publications

References 0 publications

Relationship between dispersion metric and properties of PMMA/SWNT nanocomposites

Relationship between dispersion metric and properties of PMMA/SWNT nanocomposites

Motif clustering with implications for transcription factor interactions

Average-case linear-time similar substring searching by the q -gram distance

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