Discrete Rank Research Articles

Anomalies in Vicker's microhardness of subbituminous and high volatile bituminous coals

Vickers microhardness (MH) of coal is known to be strongly correlated with coal rank. To examine coal rank and other coal quality parameters, such as organic sulfur, that might influence MH, a suite of more than 300 samples from the Penn State Coal Quality database with vitrinite Rmax < 1.1 % were examined. The data set was narrowed down to 296 coals with moisture (as-received basis) < 20 %. As MH is a parameter measured on vitrinite, vitrinite Rmax was used as the rank parameter. The Eocene Big Dirty coal (Washington state) stood out as a high MH/high-moisture coal while Hanna and Green River basin coals (Wyoming) had low atomic H/C values and K Unita Basin (Utah) coals had high H/C. Organic S did not show a correlation with MH within discrete rank ranges. With respect to vitrinite Rmax vs. MH, the Big Dirty coal and some Illinois and Iowa coals lie on the high-MH/low-Rmax side and the Pennsylvanian Tioga (West Virginia) and the Indiana Brazil Formation coals, all dominated by dull lithotypes, lie on the low-MH/high-Rmax side of the main data trend. Overall, the quadratic regression of vitrinite Rmax vs. MH yields an R2 of 0.55, indicating a significant correlation at the 95 % level.

Carbon Emissions Reduction of Neural Network by Discrete Rank Pruning

Carbon Emissions Reduction of Neural Network by Discrete Rank Pruning

Development of probabilistic assessment framework for pedestrian wind environment using Bayesian technique

This study extends the assessment framework proposed by Murakami et al. (1986) for the pedestrian wind environment to a fully probabilistic method by implementing Bayesian modeling. The method quantifies the uncertainties in constructed models based on the measured wind data and the results of the assessment. To model the probabilities of the daily maximum mean wind speed and wind direction, we employed the Weibull distribution and categorical distribution, respectively. The parameters defining the probability distributions were probabilistically modeled using Bayesian techniques. Using the wind data measured at the Meteorological Observatory of Tokyo, we demonstrated the effectiveness of the proposed method. The results showed that the wind direction probability and each parameter of the Weibull distribution could be estimated in the form of a posterior probability density function. Using the constructed models, we predicted the exceedance probability of the daily maximum instantaneous wind speed and evaluated the wind environment index (rank) in a city model. We provided a discrete rank scale in the form of a probability distribution, which enables us to quantify the evaluation uncertainties. Additionally, we clarified the effect of varying the amount of data used for the model construction. The uncertainty of the exceedance probability decreased with the amount of data. When only 1 year of data was used, some evaluation points possibly changed over three ranks. Even when 5-year observation data was used, the evaluated rank of some points varied within the range of uncertainty, thereby highlighting the importance of uncertainty quantification in the wind environment assessment.

Equidistribution and counting of orbit points for discrete rank one isometry groups of Hadamard spaces

Let $X$ be a proper, geodesically complete Hadamard space, and $\ \Gamma<\mbox{Is}(X)$ a discrete subgroup of isometries of $X$ with the fixed point of a rank one isometry of $X$ in its infinite limit set. In this paper we prove that if $\Gamma$ has non-arithmetic length spectrum, then the Ricks' Bowen-Margulis measure -- which generalizes the well-known Bowen-Margulis measure in the CAT$(-1)$ setting -- is mixing. If in addition the Ricks' Bowen-Margulis measure is finite, then we also have equidistribution of $\Gamma$-orbit points in $X$, which in particular yields an asymptotic estimate for the orbit counting function of $\Gamma$. This generalizes well-known facts for non-elementary discrete isometry groups of Hadamard manifolds with pinched negative curvature and proper CAT$(-1)$-spaces.

Analysis of the Impact of Sample Size, Attribute Variance and Within-Sample Choice Distribution on the Estimation Accuracy of Multinomial Logit Models Using Simulated Data

Literature review indicates that sample size, attribute variance and within-sample choice distribution of alternatives are important considerations in the estimation of multinomial logit (MNL) models, but their impacts on the estimation accuracy have not been systematically studied. Therefore, the objective of this paper is to provide an empirical examination to the above issues through a set of simulated discrete choice preference and rank ordered preference datasets. In this paper, the utility coefficients, alternative specific constants (ASCs), and the mean and standard deviation of the four attributes for a set of seven hypothetical alternatives are specified as a priori. Then, synthetic datasets, with varying sample size, attribute variance and within-sample choice distribution are simulated. Based on these datasets, the utility coefficients and ASCs of the specified MNLs are re-estimated and compared with the original values specified as the priori. It is found that (1) the estimation accuracy of utility parameters increases as the sample size increases; (2) the utility coefficients can be re-estimated with reasonable accuracy, but the estimates of the ASCs are confronted with much larger errors; (3) as the variances of the alternative attributes increase, the estimation accuracy improves significantly; and (4) as the distribution of chosen choices becomes more balanced across alternatives within sample datasets, the hit-ratio decreases. The results indicate that (a) under a similar setting presented in this paper, a large sample consisting of a few thousand observations (3000–4000) may be needed in order to provide reasonable estimates for utility coefficients, particularly for ASCs; (b) a larger, but realistic attribute space is preferred in the stated preference survey design; and (c) choice datasets with unbalanced “chosen” choice frequency distribution is preferred, in order to better capture the elasticity between the “perceived utility” associated with alternative’s attributes.

Robust PCA Based on Incoherence with Geometrical Interpretation.

Robust Principal Component Analysis (RPCA), which extracts low-dimensional data from high-dimensional data, can also be regarded as a source separation problem of the sparse error matrix and the low-rank matrix. Until recently, various methods have attempted to precisely predict the discrete rank function by assigning a weight to the nuclear norm. However, if the weights are not in ascending order, the algorithms will diverge and exhibit high computational complexity. Moreover, from the viewpoint of source separation, these methods overlook the fact that two components must be sufficiently different for accurate demixing. In this paper, we employ the incoherence term with convex shape, which considers that components must appear different from one another for boosting separability. Since it is intractable to directly exploit mutual incoherence defined in linear algebra, we guarantee the incoherence by indirectly making the sparse matrix lack the low-rank property by using the duality norm principle. This approach can also be associated with the null space. To analyze the results of the proposed algorithm geometrically, we measure the geodesic distance between the tangent spaces of the manifolds of two separate components. As this distance increases, the degree of dissimilarity of the two components is adequately assured; thus, separation succeeds. Furthermore, this paper is the first to provide insights into the relationship between source separation conditions and the derivatives of the nuclear norm and L1 norm. Experiments are conducted on still image separation and background subtraction to confirm the superiority of the proposed methods both qualitatively and quantitatively.

Ranking of functional data in application to worldwide PM $$_{10}$$ 10 data analysis

Ranking of functional data is important for conducting further rank-based analysis. The paper reviews several ranking methods, such as the principal component analysis/functional principal component analysis and the discrete rank method, and proposes a new method for functional data: the weighted local rank method. The proposed method allows the presence of missing values or values measured at unmatched time points. It also has physical interpretation. All the methods are compared through simulation and the proposed method is more robust in various scenarios. In real data analysis, the proposed method is applied to worldwide PM\(_{10}\) data to generate ranks, then further analysis such as nonparametric rank sum test and linear regression based on ranks are performed to produce meaningful results.

Automatic Pronunciation Scoring with Score Combination by Learning to Rank and Class-Normalized DP-Based Quantization

This paper proposes an automatic pronunciation scoring framework using learning to rank and class-normalized, dynamic-programming-based quantization. The goal is to train a model that is able to grade the pronunciation of a second language learner, such that the predicted score is as close as possible to the one given by a human teacher. Under this framework, each utterance is given a score of 1 to 5 by human raters, which is treated as a ground truth rank for the training algorithm. The corpus was rated by qualified English teachers in Taiwan (nonnative speakers). Nine phone-level scores are computed and converted into word-level scores through four conversion methods. We select the 16 best performing scores as the input features to train the learning-to-rank function. The output of the function is then quantized to a discrete rank on a 1-5 scale. The quantization is done with class normalization to alleviate the problem of data imbalance over different classes. Experimental results show that the proposed framework achieves a higher correlation to the human scores than other methods, along with higher accuracy in detecting instances of mispronunciation. We also release a new version of our nonnative corpus with human rankings.

A generalization of a theorem of Ore

Let (K,v) be a discrete rank one valued field with valuation ring Rv. Let L/K be a finite extension such that the integral closure S of Rv in L is a finitely generated Rv-module. Under a certain condition of v-regularity, we obtain some results regarding the explicit computation of Rv-bases of S, thereby generalizing similar results that had been obtained for algebraic number fields in El Fadil et al. (2012) [7]. The classical Theorem of Index of Ore is also extended to arbitrary discrete valued fields. We give a simple counter example to point out an error in the main result of Montes and Nart (1992) [12] related to the Theorem of Index and give an additional necessary and sufficient condition for this result to be valid.

On Construction of Saturated Distinguished Chains

Let υ be a Henselian valuation of arbitrary rank of a field K, and let ῡ be the (unique) extension of v to a fixed algebraic closure Kof K. For an element α e K\K, a chain α = α 0 , α 1 ,…,α r of elements of K,such that α i is of minimum degree over K with the property that ῡ(α i-1 -α i )= sup{ῡ(α i-1 -β)|[K(β): K] i -1):K]} and that α r e K , is called a saturated distinguished chain for α with respect to (K, υ). The notion of a saturated distinguished chain has been used to obtain results about the irreducible polynomials over any complete discrete rank one valued field K and to determine various arithmetic and metric invariants associated to elements of K(cf. [J. Number Theory, 52 (1995), 98–118.] and [J. Algebra, 266 (2003), 14–26]). In this paper, a method is described of constructing a saturated distinguished chain for α, and also determining explicitly some invariants associated to α, when the degree of the extension K (α)/K is not divisible by the characteristic of the residue field of υ.

Extension d’une valuation

We want to determine all the extensions of a valuation ν \nu of a field K K to a cyclic extension L L of K K , i.e. L = K ( x ) L=K(x) is the field of rational functions of x x or L = K ( θ ) L=K(\theta ) is the finite separable extension generated by a root θ \theta of an irreducible polynomial G ( x ) G(x) . In two articles from 1936, Saunders MacLane has introduced the notions of key polynomial and of augmented valuation for a given valuation μ \mu of K [ x ] K[x] , and has shown how we can recover any extension to L L of a discrete rank one valuation ν \nu of K K by a countable sequence of augmented valuations ( μ i ) i ∈ I \bigl (\mu _i\bigr ) _{i \in I} , with I ⊂ N I \subset \mathbb N . The valuation μ i \mu _i is defined by induction from the valuation μ i − 1 \mu _{i-1} , from a key polynomial ϕ i \phi _i and from the value γ i = μ ( ϕ i ) \gamma _i = \mu ( \phi _i ) . In this article we study some properties of the augmented valuations and we generalize the results of MacLane to the case of any valuation ν \nu of K K . For this we need to introduce simple admissible families of augmented valuations A = ( μ α ) α ∈ A {\mathcal A} = \bigl ( \mu _{\alpha } \bigr ) _{\alpha \in A} , where A A is not necessarily a countable set, and to define a limit key polynomial and limit augmented valuation for such families. Then, any extension μ \mu to L L of a valuation ν \nu on K K is again a limit of a family of augmented valuations. We also get a “factorization” theorem which gives a description of the values ( μ α ( f ) ) ( \mu _{\alpha } (f)) for any polynomial f f in K [ x ] K[x] .

On Chains Associated with Elements Algebraic over a Henselian Valued Field

Let v be a henselian valuation of a field K, and [Formula: see text] be the (unique) extension of v to a fixed algebraic closure [Formula: see text] of K. For an element [Formula: see text], a chain [Formula: see text] of elements of [Formula: see text] such that θi is of minimal degree over K with the property that [Formula: see text] and θm ∈ K, is called a complete distinguished chain for θ with respect to (K, v). In 1995, Popescu and Zaharescu proved the existence of a complete distinguished chain for each [Formula: see text] when (K, v) is a complete discrete rank one valued field (cf. [10]). In this paper, for a henselian valued field (K, v) of arbitrary rank, we characterize those elements [Formula: see text] for which there exists a complete distinguished chain. It is shown that a complete distinguished chain for θ gives rise to several invariants associated to θ which are same for all the K-conjugates of θ.

Valuations in fields of power series

This paper deals with valuations of fields of formal meromorphic functions and their residue fields. We explicitly describe the residue fields of the monomial valuations. We also classify all the discrete rank one valuations of fields of power series in two and three variables, according to their residue fields. We prove that all our cases are possible and give explicit constructions.

ON THE MAIN INVARIANT OF ELEMENTS ALGEBRAIC OVER A HENSELIAN VALUED FIELD

AbstractLet $v$ be a henselian valuation of a field $K$ with value group $G$, let $\bar{v}$ be the (unique) extension of $v$ to a fixed algebraic closure $\bar{K}$ of $K$ and let $(\tilde{K},\tilde{v})$ be a completion of $(K,v)$. For $\alpha\in\bar{K}\setminus K$, let $M(\alpha,K)$ denote the set $\{\bar{v}(\alpha-\beta):\beta\in\bar{K},\ [K(\beta):K] \lt [K(\alpha):K]\}$. It is known that $M(\alpha,K)$ has an upper bound in $\bar{G}$ if and only if $[K(\alpha):K]=[\tilde{K}(\alpha):\tilde{K}]$, and that the supremum of $M(\alpha,K)$, which is denoted by $\delta_{K}(\alpha)$ (usually referred to as the main invariant of $\alpha$), satisfies a principle similar to the Krasner principle. Moreover, each complete discrete rank 1 valued field $(K,v)$ has the property that $\delta_{K}(\alpha)\in M(\alpha,K)$ for every $\alpha\in\bar{K}\setminus K$. In this paper the authors give a characterization of all those henselian valued fields $(K,v)$ which have the property mentioned above.AMS 2000 Mathematics subject classification: Primary 12J10; 12J25; 13A18

On exponents and Auslander-Reiten components of irreducible lattices

Let G be a finite group, and let R be a complete discrete rank one valuation ring of characteristic zero with maximal ideal \(\max (R) = \pi R\), and residue class field \(R/\pi R\) of characteristic p > 0. The notion of the exponent of an RG-lattice L is due to J. F. Carlson and the first author [1]. In this note we use it to show that any non-projective absolutely irreducible RG-lattice L with indecomposable factor module \(\bar {L} = L/\pi L\) lies at the end of its connected component \(\Theta \) of the stable Auslander-Reiten quiver \(\Gamma _s(RG)\) of the group ring RG. Since such lattices L belong to p-blocks B with non-trivial defect groups \(\delta (B)\) we also study some relations between the order of \(\delta (B)\) and the exponent exp(L).

The Krull–Schmidt problem for modules over valuation domains

Direct sum decompositions problems for torsion-free modules of nite rank have been the subject of recent activity in the theory of modules over valuation domains (see e.g. [11]). Indeed the nal problem (Problem 26) in the text by Fuchs and Salce [6] asks if direct decompositions of nite rank modules (over a valuation domain) into indecomposable summands are unique up to isomorphism i.e. does the Krull– Schmidt Theorem hold for this class of modules? It has been known for some time that the Krull–Schmidt Theorem fails for nite-rank torsion-free modules over the discrete valuation domain Zp, the integers localized at the prime p; see [1, Exercise 2.15] for an example due essentially to M.C.R. Butler. However, there is no immediate method of extending this result to arbitrary valuation domains; indeed it is well known that if a valuation domain is Henselian, then the Krull–Schmidt Theorem does hold for torsionfree modules of nite rank (see e.g. [11, Lemma 14] or [10, Corollary 10]. In fact for discrete rank one valuation domains the two concepts are equivalent [11, Theorem 17]. One of the principal outcomes of the present work is that the Krull–Schmidt Theorem fails for a large class of non-Henselian valuation domains. It is worth remarking that this class contains many valuation domains which are not discrete and so is a farreaching generalization of [11, Theorem 17]. (We also note that Facchini has recently shown failure of Krull–Schmidt for serial modules [5].) In order to make this comment a little more precise, let us introduce the following notation: throughout, unless speci ed to the contrary, R shall denote a valuation domain, not a eld, with completion R

Discrete valuation overrings of Noetherian domains

We show that, given a chain $0=P_0\subset P_1\subset \dotsb \subset P_n$ of prime ideals in a Noetherian domain $R$, there exist a finitely generated overring $T$ of $R$ and a saturated chain of primes in $T$ contracting term by term to the given chain. We further show that there is a discrete rank $n$ valuation overring of $R$ whose primes contract to those of the given chain.

Multiplicative Independence in Function Fields

Let F be a perfect field of characteristic p. Let k be a function field in one variable over F. Let v be a discrete rank one valuation for k which is trivial on F, and let k̂ be the associated completion. Using only the separability of the extension k̂/ k, we prove a strong form of Leopoldt′s Conjecture for k.

ON THE IMMEDIATE EXTENSIONS OF A VALUED FIELD

Abstract In this paper the topological methods introduced by Kaplansky and the theory of linear compactifications are used to prove a result classifying the maximal immediate extensions of a valued field. Results on the existence of a complete discrete rank n valued field of characteristic 0 with prescribed residue class field of characteristic p > 0 are discussed. By applying results of Endler and Ribenboim the existence of a valued field of characteristic 0 and having prescribed residue field of characteristic p > 0 when the value group has finite rank but need not be discrete is demonstrated.

Auslander-reiten quivers of schurian orders

We shall consider orders ⋀ over a complete discrete rank one valuation ring R in a split full matrix ring containing a complete sex of primitive orthogonal idempot ents. In case s of finite lattice type and R is the power series ring in one variable over its residexe class field k , we give a description of its index composable lattices and its Auslander-Reiten quiver in terms of representations of partially ordered sets. By a model theoretic argument, this implies a description of all indecomposable lattices if R is the completion of the ring of integers in an algebraic number field at all but possibly a finite number of primes.